Runtime Complexity: TRS pair #487109686

details

property	value
status	complete
benchmark	13.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n149.star.cs.uiowa.edu
space	Various_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	294.296 seconds
cpu usage	1090.82
user time	1080.89
system time	9.92973
max virtual memory	1.9607704E7
max residence set size	1.5058E7

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 0 ms]
(2) CpxTRS
    (3) NonCtorToCtorProof [UPPER BOUND(ID), 0 ms]
    (4) CpxRelTRS
    (5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 354 ms]
    (18) BOUNDS(1, n^2)
(19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(20) TRS for Loop Detection
    (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) 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)))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(O(x), y) -> O(*(x, y))
   *(I(x), y) -> +(O(*(x, y)), y)
   -(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))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of O: O, +, -, *
The following defined symbols can occur below the 0th argument of +: O, +, *

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
+(O(x), O(y)) -> O(+(x, y))
+(I(x), O(y)) -> I(+(x, y))
*(O(x), y) -> O(*(x, y))
-(O(x), O(y)) -> O(-(x, y))
-(O(x), I(y)) -> I(-(-(x, y), I(1)))
-(I(x), O(y)) -> I(-(x, y))

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

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


The TRS R consists of the following rules:

   O(0) -> 0
   +(0, x) -> x
   +(x, 0) -> x
   +(O(x), I(y)) -> I(+(x, y))
   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(I(x), y) -> +(O(*(x, y)), y)
   -(x, 0) -> x
   -(0, x) -> 0
   -(I(x), I(y)) -> O(-(x, y))

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