Runtime Complexity: TRS pair #487109688

details

property	value
status	complete
benchmark	14.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n141.star.cs.uiowa.edu
space	Various_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	291.586 seconds
cpu usage	628.82
user time	624.897
system time	3.92243
max virtual memory	1.9747076E7
max residence set size	6483424.0

stage attributes

key	value
starexec-result	WORST_CASE(Omega(n^1), O(n^2))

output

WORST_CASE(Omega(n^1), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^2).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 14 ms]
(2) CpxTRS
    (3) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
    (4) CpxRelTRS
    (5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) CpxTrsToCdtProof [UPPER BOUND(ID), 5 ms]
    (8) CdtProblem
    (9) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (16) CdtProblem
    (17) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
    (19) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 141 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms]
    (24) CdtProblem
    (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 49 ms]
    (26) CdtProblem
    (27) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 45 ms]
    (28) CdtProblem
    (29) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 435 ms]
    (30) CdtProblem
    (31) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 414 ms]
    (32) CdtProblem
    (33) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) BOUNDS(1, 1)
(35) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(36) TRS for Loop Detection
    (37) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (38) BEST
        (39) proven lower bound
            (40) LowerBoundPropagationProof [FINISHED, 0 ms]
            (41) BOUNDS(n^1, INF)
        (42) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   O(0) -> 0
   +(0, x) -> x
   +(x, 0) -> x
   +(O(x), O(y)) -> O(+(x, y))
   +(O(x), I(y)) -> I(+(x, y))
   +(I(x), O(y)) -> I(+(x, y))
   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
   +(x, +(y, z)) -> +(+(x, y), z)
   -(x, 0) -> x
   -(0, x) -> 0
   -(O(x), O(y)) -> O(-(x, y))
   -(O(x), I(y)) -> I(-(-(x, y), I(1)))
   -(I(x), O(y)) -> I(-(x, y))
   -(I(x), I(y)) -> O(-(x, y))
   not(true) -> false
   not(false) -> true
   and(x, true) -> x
   and(x, false) -> false
   if(true, x, y) -> x
   if(false, x, y) -> y
   ge(O(x), O(y)) -> ge(x, y)
   ge(O(x), I(y)) -> not(ge(y, x))
   ge(I(x), O(y)) -> ge(x, y)
   ge(I(x), I(y)) -> ge(x, y)
   ge(x, 0) -> true
   ge(0, O(x)) -> ge(0, x)
   ge(0, I(x)) -> false
   Log'(0) -> 0
   Log'(I(x)) -> +(Log'(x), I(0))
   Log'(O(x)) -> if(ge(x, I(0)), +(Log'(x), I(0)), 0)
   Log(x) -> -(Log'(x), I(0))
   Val(L(x)) -> x
   Val(N(x, l, r)) -> x
   Min(L(x)) -> x
   Min(N(x, l, r)) -> Min(l)
   Max(L(x)) -> x
   Max(N(x, l, r)) -> Max(r)
   BS(L(x)) -> true
   BS(N(x, l, r)) -> and(and(ge(x, Max(l)), ge(Min(r), x)), and(BS(l), BS(r)))

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