Derivational Complexity: TRS Innermost pair #487105824

details

property	value
status	complete
benchmark	x02.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n137.star.cs.uiowa.edu
space	Secret_07_SRS

run statistics

property	value
solver	AProVE
configuration	rcdcRelativeAlsoLower
runtime (wallclock)	291.504 seconds
cpu usage	371.252
user time	360.87
system time	10.3811
max virtual memory	3.7318056E7
max residence set size	1.44155E7

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 58 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^1)), 111 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 98 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 97 ms]
    (20) CdtProblem
    (21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) BOUNDS(1, 1)
(23) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxRelTRS
    (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 12 ms]
    (26) typed CpxTrs
    (27) OrderProof [LOWER BOUND(ID), 0 ms]
    (28) typed CpxTrs
    (29) RewriteLemmaProof [LOWER BOUND(ID), 1255 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 503 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 106 ms]
            (40) BOUNDS(1, INF)


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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(b(x1)) -> c(d(x1))
   d(d(x1)) -> b(e(x1))
   b(x1) -> d(c(x1))
   d(x1) -> x1
   e(c(x1)) -> d(a(x1))
   a(x1) -> e(d(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(c(x_1)) -> c(encArg(x_1))
   encArg(cons_a(x_1)) -> a(encArg(x_1))
   encArg(cons_d(x_1)) -> d(encArg(x_1))
   encArg(cons_b(x_1)) -> b(encArg(x_1))
   encArg(cons_e(x_1)) -> e(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))
   encode_d(x_1) -> d(encArg(x_1))
   encode_e(x_1) -> e(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(b(x1)) -> c(d(x1))
   d(d(x1)) -> b(e(x1))
   b(x1) -> d(c(x1))
   d(x1) -> x1
   e(c(x1)) -> d(a(x1))
   a(x1) -> e(d(x1))

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