Derivational Complexity: TRS Innermost pair #487107966

details

property	value
status	complete
benchmark	01.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n149.star.cs.uiowa.edu
space	Der95

run statistics

property	value
solver	AProVE
configuration	rcdcRelativeAlsoLower
runtime (wallclock)	5.13177 seconds
cpu usage	14.6592
user time	13.3271
system time	1.33201
max virtual memory	1.888214E7
max residence set size	4022688.0

stage attributes

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

output

WORST_CASE(Omega(n^1), O(n^1))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) DCpxTrs
(1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxRelTRS
(3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 194 ms]
(4) CpxRelTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 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)), 93 ms]
    (16) CdtProblem
    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) BOUNDS(1, 1)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 230 ms]
    (26) proven lower bound
    (27) LowerBoundPropagationProof [FINISHED, 0 ms]
    (28) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   \(x, x) -> e
   /(x, x) -> e
   .(e, x) -> x
   .(x, e) -> x
   \(e, x) -> x
   /(x, e) -> x
   .(x, \(x, y)) -> y
   .(/(y, x), x) -> y
   \(x, .(x, y)) -> y
   /(.(y, x), x) -> y
   /(x, \(y, x)) -> y
   \(/(x, y), x) -> y

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(e) -> e
   encArg(cons_\(x_1, x_2)) -> \(encArg(x_1), encArg(x_2))
   encArg(cons_/(x_1, x_2)) -> /(encArg(x_1), encArg(x_2))
   encArg(cons_.(x_1, x_2)) -> .(encArg(x_1), encArg(x_2))
   encode_\(x_1, x_2) -> \(encArg(x_1), encArg(x_2))
   encode_e -> e
   encode_/(x_1, x_2) -> /(encArg(x_1), encArg(x_2))
   encode_.(x_1, x_2) -> .(encArg(x_1), encArg(x_2))


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

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


The TRS R consists of the following rules:

   \(x, x) -> e
   /(x, x) -> e
   .(e, x) -> x
   .(x, e) -> x
   \(e, x) -> x
   /(x, e) -> x
   .(x, \(x, y)) -> y
   .(/(y, x), x) -> y
   \(x, .(x, y)) -> y
   /(.(y, x), x) -> y
   /(x, \(y, x)) -> y
   \(/(x, y), x) -> y

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

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job log

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actions all output return to Derivational Complexity: TRS Innermost