Derivational Complexity: TRS pair #487104054

details

property	value
status	complete
benchmark	Ex25_Luc06_GM.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	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	rcdcRelativeAlsoLower
runtime (wallclock)	291.511 seconds
cpu usage	341.048
user time	337.972
system time	3.07616
max virtual memory	3.7582956E7
max residence set size	5887280.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 Derivational Complexity (full) 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), 55 ms]
(4) CpxRelTRS
(5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
    (7) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (8) CdtProblem
    (9) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (10) CdtProblem
    (11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (14) CdtProblem
    (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 198 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 172 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 168 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 120 ms]
    (24) CdtProblem
    (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 126 ms]
    (26) CdtProblem
    (27) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) BOUNDS(1, 1)
(29) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(30) TRS for Loop Detection
    (31) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) TRS for Loop Detection


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


The TRS R consists of the following rules:

   a__f(f(X)) -> a__c(f(g(f(X))))
   a__c(X) -> d(X)
   a__h(X) -> a__c(d(X))
   mark(f(X)) -> a__f(mark(X))
   mark(c(X)) -> a__c(X)
   mark(h(X)) -> a__h(mark(X))
   mark(g(X)) -> g(X)
   mark(d(X)) -> d(X)
   a__f(X) -> f(X)
   a__c(X) -> c(X)
   a__h(X) -> h(X)

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

(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(f(x_1)) -> f(encArg(x_1))
   encArg(g(x_1)) -> g(encArg(x_1))
   encArg(d(x_1)) -> d(encArg(x_1))
   encArg(c(x_1)) -> c(encArg(x_1))
   encArg(h(x_1)) -> h(encArg(x_1))
   encArg(cons_a__f(x_1)) -> a__f(encArg(x_1))
   encArg(cons_a__c(x_1)) -> a__c(encArg(x_1))
   encArg(cons_a__h(x_1)) -> a__h(encArg(x_1))
   encArg(cons_mark(x_1)) -> mark(encArg(x_1))
   encode_a__f(x_1) -> a__f(encArg(x_1))
   encode_f(x_1) -> f(encArg(x_1))
   encode_a__c(x_1) -> a__c(encArg(x_1))
   encode_g(x_1) -> g(encArg(x_1))
   encode_d(x_1) -> d(encArg(x_1))
   encode_a__h(x_1) -> a__h(encArg(x_1))
   encode_mark(x_1) -> mark(encArg(x_1))
   encode_c(x_1) -> c(encArg(x_1))
   encode_h(x_1) -> h(encArg(x_1))


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

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