Runtime Complexity: TRS Innermost pair #487112764

details

property	value
status	complete
benchmark	2.12.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n145.star.cs.uiowa.edu
space	SK90

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	6.79061 seconds
cpu usage	23.1717
user time	21.4937
system time	1.67802
max virtual memory	1.9145508E7
max residence set size	4140716.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 66 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 79 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 523 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 271 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 1719 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
            (28) proven lower bound
            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
            (30) 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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(y))

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   +(p(z0), z1) -> p(+(z0, z1))
   minus(0) -> 0
   minus(s(z0)) -> p(minus(z0))
   minus(p(z0)) -> s(minus(z0))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   *(p(z0), z1) -> +(*(z0, z1), minus(z1))
Tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(0) -> c3
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(0, z0) -> c6
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
S tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(0) -> c3
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(0, z0) -> c6
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))

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