Runtime Complexity: TRS Innermost pair #487112904

details

property	value
status	complete
benchmark	division.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n138.star.cs.uiowa.edu
space	Rubio_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	291.514 seconds
cpu usage	314.311
user time	311.55
system time	2.76139
max virtual memory	1.9075592E7
max residence set size	5170172.0

stage attributes

key	value
starexec-result	WORST_CASE(Omega(n^1), O(n^3))

output

WORST_CASE(Omega(n^1), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 251 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 99 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 724 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 250 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 330 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^3)
(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


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   minus(0, Y) -> 0
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
   ifMinus(true, s(X), Y) -> 0
   ifMinus(false, s(X), Y) -> s(minus(X, Y))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   le(0, Y) -> true [1]
   le(s(X), 0) -> false [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   minus(0, Y) -> 0 [1]
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y) [1]
   ifMinus(true, s(X), Y) -> 0 [1]
   ifMinus(false, s(X), Y) -> s(minus(X, Y)) [1]
   quot(0, s(Y)) -> 0 [1]
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))

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