Runti Compl Inner Rewri 22807 pair #381904374

details

property	value
status	complete
benchmark	subtrees.raml.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n040.star.cs.uiowa.edu
space	raML

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	291.4735322 seconds
cpu usage	310.423477686
max memory	5.315465216E9

stage attributes

key	value
output-size	17726
starexec-result	WORST_CASE(Omega(n^1), O(n^2))

output

/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(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)), 119 ms]
    (6) CdtProblem
    (7) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 118 ms]
    (10) CdtProblem
    (11) CdtKnowledgeProof [FINISHED, 0 ms]
    (12) BOUNDS(1, 1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   append(@l1, @l2) -> append#1(@l1, @l2)
   append#1(::(@x, @xs), @l2) -> ::(@x, append(@xs, @l2))
   append#1(nil, @l2) -> @l2
   subtrees(@t) -> subtrees#1(@t)
   subtrees#1(leaf) -> nil
   subtrees#1(node(@x, @t1, @t2)) -> subtrees#2(subtrees(@t1), @t1, @t2, @x)
   subtrees#2(@l1, @t1, @t2, @x) -> subtrees#3(subtrees(@t2), @l1, @t1, @t2, @x)
   subtrees#3(@l2, @l1, @t1, @t2, @x) -> ::(node(@x, @t1, @t2), append(@l1, @l2))

S is empty.
Rewrite Strategy: INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   append(z0, z1) -> append#1(z0, z1)
   append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2))
   append#1(nil, z0) -> z0
   subtrees(z0) -> subtrees#1(z0)
   subtrees#1(leaf) -> nil
   subtrees#1(node(z0, z1, z2)) -> subtrees#2(subtrees(z1), z1, z2, z0)
   subtrees#2(z0, z1, z2, z3) -> subtrees#3(subtrees(z2), z0, z1, z2, z3)
   subtrees#3(z0, z1, z2, z3, z4) -> ::(node(z4, z2, z3), append(z1, z0))
Tuples:
   APPEND(z0, z1) -> c(APPEND#1(z0, z1))
   APPEND#1(::(z0, z1), z2) -> c1(APPEND(z1, z2))
   APPEND#1(nil, z0) -> c2
   SUBTREES(z0) -> c3(SUBTREES#1(z0))
   SUBTREES#1(leaf) -> c4
   SUBTREES#1(node(z0, z1, z2)) -> c5(SUBTREES#2(subtrees(z1), z1, z2, z0), SUBTREES(z1))
   SUBTREES#2(z0, z1, z2, z3) -> c6(SUBTREES#3(subtrees(z2), z0, z1, z2, z3), SUBTREES(z2))
   SUBTREES#3(z0, z1, z2, z3, z4) -> c7(APPEND(z1, z0))
S tuples:
   APPEND(z0, z1) -> c(APPEND#1(z0, z1))
   APPEND#1(::(z0, z1), z2) -> c1(APPEND(z1, z2))
   APPEND#1(nil, z0) -> c2
   SUBTREES(z0) -> c3(SUBTREES#1(z0))
   SUBTREES#1(leaf) -> c4
   SUBTREES#1(node(z0, z1, z2)) -> c5(SUBTREES#2(subtrees(z1), z1, z2, z0), SUBTREES(z1))
   SUBTREES#2(z0, z1, z2, z3) -> c6(SUBTREES#3(subtrees(z2), z0, z1, z2, z3), SUBTREES(z2))
   SUBTREES#3(z0, z1, z2, z3, z4) -> c7(APPEND(z1, z0))
K tuples:none
Defined Rule Symbols:   append_2, append#1_2, subtrees_1, subtrees#1_1, subtrees#2_4, subtrees#3_5

Defined Pair Symbols:   APPEND_2, APPEND#1_2, SUBTREES_1, SUBTREES#1_1, SUBTREES#2_4, SUBTREES#3_5

Compound Symbols:   c_1, c1_1, c2, c3_1, c4, c5_2, c6_2, c7_1

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