Runtime Complexity: TRS Innermost pair #487111664

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	n147.star.cs.uiowa.edu
space	raML

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	291.536 seconds
cpu usage	314.65
user time	312.691
system time	1.95881
max virtual memory	1.8879708E7
max residence set size	5226900.0

stage attributes

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

output

WORST_CASE(Omega(n^1), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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)), 73 ms]
    (6) CdtProblem
    (7) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 107 ms]
    (10) CdtProblem
    (11) CdtKnowledgeProof [FINISHED, 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxTRS
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) typed CpxTrs
    (19) OrderProof [LOWER BOUND(ID), 0 ms]
    (20) typed CpxTrs
    (21) RewriteLemmaProof [LOWER BOUND(ID), 212 ms]
    (22) BEST
        (23) proven lower bound
            (24) LowerBoundPropagationProof [FINISHED, 0 ms]
            (25) BOUNDS(n^1, INF)
        (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
            (28) typed CpxTrs


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

(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
----------------------------------------

(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

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