WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 284 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), 73 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
        (16) BOUNDS(1, INF)


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

(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:

   minus_active(0, y) -> 0
   mark(0) -> 0
   minus_active(s(x), s(y)) -> minus_active(x, y)
   mark(s(x)) -> s(mark(x))
   ge_active(x, 0) -> true
   mark(minus(x, y)) -> minus_active(x, y)
   ge_active(0, s(y)) -> false
   mark(ge(x, y)) -> ge_active(x, y)
   ge_active(s(x), s(y)) -> ge_active(x, y)
   mark(div(x, y)) -> div_active(mark(x), y)
   div_active(0, s(y)) -> 0
   mark(if(x, y, z)) -> if_active(mark(x), y, z)
   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0)
   if_active(true, x, y) -> mark(x)
   minus_active(x, y) -> minus(x, y)
   if_active(false, x, y) -> mark(y)
   ge_active(x, y) -> ge(x, y)
   if_active(x, y, z) -> if(x, y, z)
   div_active(x, y) -> div(x, y)

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:

   minus_active(0', y) -> 0'
   mark(0') -> 0'
   minus_active(s(x), s(y)) -> minus_active(x, y)
   mark(s(x)) -> s(mark(x))
   ge_active(x, 0') -> true
   mark(minus(x, y)) -> minus_active(x, y)
   ge_active(0', s(y)) -> false
   mark(ge(x, y)) -> ge_active(x, y)
   ge_active(s(x), s(y)) -> ge_active(x, y)
   mark(div(x, y)) -> div_active(mark(x), y)
   div_active(0', s(y)) -> 0'
   mark(if(x, y, z)) -> if_active(mark(x), y, z)
   div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
   if_active(true, x, y) -> mark(x)
   minus_active(x, y) -> minus(x, y)
   if_active(false, x, y) -> mark(y)
   ge_active(x, y) -> ge(x, y)
   if_active(x, y, z) -> if(x, y, z)
   div_active(x, y) -> div(x, y)

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

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

(4)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus_active, mark, ge_active

They will be analysed ascendingly in the following order:
minus_active < mark
ge_active < mark

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

(6)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if


Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x))


The following defined symbols remain to be analysed:
minus_active, mark, ge_active

They will be analysed ascendingly in the following order:
minus_active < mark
ge_active < mark

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(0), rt in Omega(1 + n4_0)

Induction Base:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(0), gen_0':s:true:minus:false:ge:div:if2_0(0)) ->_R^Omega(1)
0'

Induction Step:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(+(n4_0, 1)), gen_0':s:true:minus:false:ge:div:if2_0(+(n4_0, 1))) ->_R^Omega(1)
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) ->_IH
gen_0':s:true:minus:false:ge:div:if2_0(0)

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:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if


Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x))


The following defined symbols remain to be analysed:
minus_active, mark, ge_active

They will be analysed ascendingly in the following order:
minus_active < mark
ge_active < mark

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if


Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x))


The following defined symbols remain to be analysed:
ge_active, mark

They will be analysed ascendingly in the following order:
ge_active < mark

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n700_0), gen_0':s:true:minus:false:ge:div:if2_0(n700_0)) -> true, rt in Omega(1 + n700_0)

Induction Base:
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(0), gen_0':s:true:minus:false:ge:div:if2_0(0)) ->_R^Omega(1)
true

Induction Step:
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(+(n700_0, 1)), gen_0':s:true:minus:false:ge:div:if2_0(+(n700_0, 1))) ->_R^Omega(1)
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n700_0), gen_0':s:true:minus:false:ge:div:if2_0(n700_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:
Innermost TRS:
Rules:
minus_active(0', y) -> 0'
mark(0') -> 0'
minus_active(s(x), s(y)) -> minus_active(x, y)
mark(s(x)) -> s(mark(x))
ge_active(x, 0') -> true
mark(minus(x, y)) -> minus_active(x, y)
ge_active(0', s(y)) -> false
mark(ge(x, y)) -> ge_active(x, y)
ge_active(s(x), s(y)) -> ge_active(x, y)
mark(div(x, y)) -> div_active(mark(x), y)
div_active(0', s(y)) -> 0'
mark(if(x, y, z)) -> if_active(mark(x), y, z)
div_active(s(x), s(y)) -> if_active(ge_active(x, y), s(div(minus(x, y), s(y))), 0')
if_active(true, x, y) -> mark(x)
minus_active(x, y) -> minus(x, y)
if_active(false, x, y) -> mark(y)
ge_active(x, y) -> ge(x, y)
if_active(x, y, z) -> if(x, y, z)
div_active(x, y) -> div(x, y)

Types:
minus_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
0' :: 0':s:true:minus:false:ge:div:if
mark :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
s :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
ge_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
true :: 0':s:true:minus:false:ge:div:if
minus :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
false :: 0':s:true:minus:false:ge:div:if
ge :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
div_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
if_active :: 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if -> 0':s:true:minus:false:ge:div:if
hole_0':s:true:minus:false:ge:div:if1_0 :: 0':s:true:minus:false:ge:div:if
gen_0':s:true:minus:false:ge:div:if2_0 :: Nat -> 0':s:true:minus:false:ge:div:if


Lemmas:
minus_active(gen_0':s:true:minus:false:ge:div:if2_0(n4_0), gen_0':s:true:minus:false:ge:div:if2_0(n4_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(0), rt in Omega(1 + n4_0)
ge_active(gen_0':s:true:minus:false:ge:div:if2_0(n700_0), gen_0':s:true:minus:false:ge:div:if2_0(n700_0)) -> true, rt in Omega(1 + n700_0)


Generator Equations:
gen_0':s:true:minus:false:ge:div:if2_0(0) <=> 0'
gen_0':s:true:minus:false:ge:div:if2_0(+(x, 1)) <=> s(gen_0':s:true:minus:false:ge:div:if2_0(x))


The following defined symbols remain to be analysed:
mark
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_0':s:true:minus:false:ge:div:if2_0(n1492_0)) -> gen_0':s:true:minus:false:ge:div:if2_0(n1492_0), rt in Omega(1 + n1492_0)

Induction Base:
mark(gen_0':s:true:minus:false:ge:div:if2_0(0)) ->_R^Omega(1)
0'

Induction Step:
mark(gen_0':s:true:minus:false:ge:div:if2_0(+(n1492_0, 1))) ->_R^Omega(1)
s(mark(gen_0':s:true:minus:false:ge:div:if2_0(n1492_0))) ->_IH
s(gen_0':s:true:minus:false:ge:div:if2_0(c1493_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(16)
BOUNDS(1, INF)