Derivational Complexity: TRS Innermost pair #487104552

details

property	value
status	complete
benchmark	size-12-alpha-3-num-110.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n143.star.cs.uiowa.edu
space	Waldmann_07_size12

run statistics

property	value
solver	AProVE
configuration	rcdcRelativeAlsoLower
runtime (wallclock)	291.548 seconds
cpu usage	586.831
user time	576.384
system time	10.4475
max virtual memory	1.8681608E7
max residence set size	1.5316048E7

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 64 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (12) CdtProblem
    (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 63 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 45 ms]
    (18) CdtProblem
    (19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) BOUNDS(1, 1)
(21) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(22) TRS for Loop Detection
    (23) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) TRS for Loop Detection


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


The TRS R consists of the following rules:

   a(x1) -> x1
   a(a(b(x1))) -> b(b(a(a(x1))))
   c(b(x1)) -> c(a(x1))

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID))
The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem:

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))


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

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


The TRS R consists of the following rules:

   a(x1) -> x1
   a(a(b(x1))) -> b(b(a(a(x1))))
   c(b(x1)) -> c(a(x1))

The (relative) TRS S consists of the following rules:

   encArg(b(x_1)) -> b(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_c(x_1)) -> c(encArg(x_1))
   encode_a(x_1) -> a(encArg(x_1))
   encode_b(x_1) -> b(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))

Rewrite Strategy: INNERMOST
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(3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).

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