Runtime Complexity: TRS Innermost pair #487112684

details

property	value
status	complete
benchmark	lte.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	Frederiksen_Glenstrup

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	4.52938 seconds
cpu usage	14.4375
user time	13.3095
system time	1.12798
max virtual memory	1.8845508E7
max residence set size	3418392.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 42 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) CompleteCoflocoProof [FINISHED, 244 ms]
    (12) BOUNDS(1, n^1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRelTRS
    (15) SlicingProof [LOWER BOUND(ID), 0 ms]
    (16) CpxRelTRS
    (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), 247 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), 3 ms]
            (28) BOUNDS(1, INF)


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

(0)
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:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs)
   lte(Cons(x, xs), Nil) -> False
   even(Cons(x, Nil)) -> False
   even(Cons(x', Cons(x, xs))) -> even(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(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:

   lte(Cons(x', xs'), Cons(x, xs)) -> lte(xs', xs)
   lte(Cons(x, xs), Nil) -> False
   even(Cons(x, Nil)) -> False
   even(Cons(x', Cons(x, xs))) -> even(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lte(Nil, y) -> True
   even(Nil) -> True
   goal(x, y) -> and(lte(x, y), even(x))

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS

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