Runtime Complexity: TRS Innermost pair #487111786

details

property	value
status	complete
benchmark	22.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n140.star.cs.uiowa.edu
space	Various_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	291.512 seconds
cpu usage	310.658
user time	308.085
system time	2.57238
max virtual memory	1.9010496E7
max residence set size	5230696.0

stage attributes

key	value
starexec-result	WORST_CASE(Omega(n^1), O(n^3))

output

WORST_CASE(Omega(n^1), 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, 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)), 25 ms]
    (8) CdtProblem
    (9) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (16) CdtProblem
    (17) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 8 ms]
    (20) CdtProblem
    (21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) BOUNDS(1, 1)
(23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 0 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 279 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   f(x, 0) -> s(0)
   f(s(x), s(y)) -> s(f(x, y))
   g(0, x) -> g(f(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:
   f(z0, 0) -> s(0)
   f(s(z0), s(z1)) -> s(f(z0, z1))
   g(0, z0) -> g(f(z0, z0), z0)
Tuples:
   F(z0, 0) -> c
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
S tuples:
   F(z0, 0) -> c
   F(s(z0), s(z1)) -> c1(F(z0, z1))
   G(0, z0) -> c2(G(f(z0, z0), z0), F(z0, z0))
K tuples:none
Defined Rule Symbols:   f_2, g_2

Defined Pair Symbols:   F_2, G_2

Compound Symbols:   c, c1_1, c2_2


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(z0, 0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

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