Runtime Complexity: TRS Innermost pair #487112810

details

property	value
status	complete
benchmark	2.13.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n148.star.cs.uiowa.edu
space	SK90

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	4.89026 seconds
cpu usage	15.4289
user time	13.9353
system time	1.49364
max virtual memory	3.7112188E7
max residence set size	4052876.0

stage attributes

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

output

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(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)), 36 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 7 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), 252 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), 101 ms]
            (26) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(s(x), y) -> s(+(x, y))
   double(x) -> +(x, 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:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   double(z0) -> +(z0, z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   +(s(z0), z1) -> s(+(z0, z1))
Tuples:
   DOUBLE(0) -> c
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, 0) -> c3
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
S tuples:
   DOUBLE(0) -> c
   DOUBLE(s(z0)) -> c1(DOUBLE(z0))
   DOUBLE(z0) -> c2(+'(z0, z0))
   +'(z0, 0) -> c3
   +'(z0, s(z1)) -> c4(+'(z0, z1))
   +'(s(z0), z1) -> c5(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   double_1, +_2

Defined Pair Symbols:   DOUBLE_1, +'_2

Compound Symbols:   c, c1_1, c2_1, c3, c4_1, c5_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   DOUBLE(0) -> c
   +'(z0, 0) -> c3

popout

output may be truncated. 'popout' for the full output.

job log

popout

actions all output return to Runtime Complexity: TRS Innermost