/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 264 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), 59 ms]
        (14) 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:

   minus(x, y) -> if(gt(x, y), x, y)
   if(true, x, y) -> s(minus(p(x), y))
   if(false, x, y) -> 0
   p(0) -> 0
   p(s(x)) -> x
   ge(x, 0) -> true
   ge(0, s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   div(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0), x, y)
   if1(false, x, y) -> 0
   if2(true, x, y) -> s(div(minus(x, y), y))
   if2(false, x, y) -> 0

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:

   minus(x, y) -> if(gt(x, y), x, y)
   if(true, x, y) -> s(minus(p(x), y))
   if(false, x, y) -> 0'
   p(0') -> 0'
   p(s(x)) -> x
   ge(x, 0') -> true
   ge(0', s(x)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   gt(0', y) -> false
   gt(s(x), 0') -> true
   gt(s(x), s(y)) -> gt(x, y)
   div(x, y) -> if1(ge(x, y), x, y)
   if1(true, x, y) -> if2(gt(y, 0'), x, y)
   if1(false, x, y) -> 0'
   if2(true, x, y) -> s(div(minus(x, y), y))
   if2(false, x, y) -> 0'

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

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

(4)
Obligation:
TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, gt, ge, div

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

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

(6)
Obligation:
TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gt, minus, ge, div

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> false, rt in Omega(1 + n5_0)

Induction Base:
gt(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_s:0'3_0(+(n5_0, 1)), gen_s:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
gt(gen_s:0'3_0(n5_0), gen_s:0'3_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:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gt, minus, ge, div

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> false, rt in Omega(1 + n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
minus, ge, div

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_s:0'3_0(n335_0), gen_s:0'3_0(n335_0)) -> true, rt in Omega(1 + n335_0)

Induction Base:
ge(gen_s:0'3_0(0), gen_s:0'3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_s:0'3_0(+(n335_0, 1)), gen_s:0'3_0(+(n335_0, 1))) ->_R^Omega(1)
ge(gen_s:0'3_0(n335_0), gen_s:0'3_0(n335_0)) ->_IH
true

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

(14)
Obligation:
TRS:
Rules:
minus(x, y) -> if(gt(x, y), x, y)
if(true, x, y) -> s(minus(p(x), y))
if(false, x, y) -> 0'
p(0') -> 0'
p(s(x)) -> x
ge(x, 0') -> true
ge(0', s(x)) -> false
ge(s(x), s(y)) -> ge(x, y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
div(x, y) -> if1(ge(x, y), x, y)
if1(true, x, y) -> if2(gt(y, 0'), x, y)
if1(false, x, y) -> 0'
if2(true, x, y) -> s(div(minus(x, y), y))
if2(false, x, y) -> 0'

Types:
minus :: s:0' -> s:0' -> s:0'
if :: true:false -> s:0' -> s:0' -> s:0'
gt :: s:0' -> s:0' -> true:false
true :: true:false
s :: s:0' -> s:0'
p :: s:0' -> s:0'
false :: true:false
0' :: s:0'
ge :: s:0' -> s:0' -> true:false
div :: s:0' -> s:0' -> s:0'
if1 :: true:false -> s:0' -> s:0' -> s:0'
if2 :: true:false -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_true:false2_0 :: true:false
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'3_0(n5_0), gen_s:0'3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
ge(gen_s:0'3_0(n335_0), gen_s:0'3_0(n335_0)) -> true, rt in Omega(1 + n335_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
div