Runtime Complexity: TRS Innermost pair #487112504

details

property	value
status	complete
benchmark	#3.41.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	AG01

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	5.11785 seconds
cpu usage	16.8631
user time	15.4254
system time	1.43762
max virtual memory	1.9136764E7
max residence set size	4104488.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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

(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:

   p(s(x)) -> x
   fac(0) -> s(0)
   fac(s(x)) -> times(s(x), fac(p(s(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:
   p(s(z0)) -> z0
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))
Tuples:
   P(s(z0)) -> c
   FAC(0) -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
S tuples:
   P(s(z0)) -> c
   FAC(0) -> c1
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:   p_1, fac_1

Defined Pair Symbols:   P_1, FAC_1

Compound Symbols:   c, c1, c2_2


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   FAC(0) -> c1
   P(s(z0)) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fac(0) -> s(0)
   fac(s(z0)) -> times(s(z0), fac(p(s(z0))))
Tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))
S tuples:
   FAC(s(z0)) -> c2(FAC(p(s(z0))), P(s(z0)))

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