Runtime Complexity: TRS pair #487109692

details

property	value
status	complete
benchmark	12.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n144.star.cs.uiowa.edu
space	Various_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	291.74 seconds
cpu usage	1096.02
user time	1085.87
system time	10.148
max virtual memory	2.0286492E7
max residence set size	1.465542E7

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 2 ms]
    (14) CpxTypedWeightedCompleteTrs
    (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 1 ms]
    (16) CpxRNTS
    (17) CompleteCoflocoProof [FINISHED, 289 ms]
    (18) BOUNDS(1, n^2)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 1051 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 37 ms]
            (32) BOUNDS(1, INF)


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

(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)

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

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

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

(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)

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

(3) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system

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