Runtime Complexity: TRS Innermost pair #487112470

details

property	value
status	complete
benchmark	#3.12.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n151.star.cs.uiowa.edu
space	AG01

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	3.75534 seconds
cpu usage	11.8889
user time	11.216
system time	0.672966
max virtual memory	1.9017864E7
max residence set size	1826172.0

stage attributes

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

output

WORST_CASE(Omega(n^3), O(n^3))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, n^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 25 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 74 ms]
    (12) CdtProblem
    (13) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) BOUNDS(1, 1)
(15) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTRS
    (17) SlicingProof [LOWER BOUND(ID), 0 ms]
    (18) CpxTRS
    (19) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) typed CpxTrs
    (21) OrderProof [LOWER BOUND(ID), 0 ms]
    (22) typed CpxTrs
    (23) RewriteLemmaProof [LOWER BOUND(ID), 284 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 16 ms]
            (30) BEST
                (31) proven lower bound
                    (32) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (33) BOUNDS(n^2, INF)
                (34) typed CpxTrs
                    (35) RewriteLemmaProof [LOWER BOUND(ID), 12 ms]
                    (36) proven lower bound
                    (37) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (38) BOUNDS(n^3, INF)


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

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


The TRS R consists of the following rules:

   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   reverse(nil) -> nil
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
   shuffle(nil) -> nil
   shuffle(add(z0, z1)) -> add(z0, shuffle(reverse(z1)))
Tuples:
   APP(nil, z0) -> c
   APP(add(z0, z1), z2) -> c1(APP(z1, z2))
   REVERSE(nil) -> c2
   REVERSE(add(z0, z1)) -> c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(nil) -> c4
   SHUFFLE(add(z0, z1)) -> c5(SHUFFLE(reverse(z1)), REVERSE(z1))
S tuples:
   APP(nil, z0) -> c
   APP(add(z0, z1), z2) -> c1(APP(z1, z2))
   REVERSE(nil) -> c2
   REVERSE(add(z0, z1)) -> c3(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(nil) -> c4
   SHUFFLE(add(z0, z1)) -> c5(SHUFFLE(reverse(z1)), REVERSE(z1))
K tuples:none
Defined Rule Symbols:   app_2, reverse_1, shuffle_1

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