Runtime Complexity: TRS pair #487109984

details

property	value
status	complete
benchmark	LengthOfFiniteLists_nokinds_C.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	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	291.687 seconds
cpu usage	1107.82
user time	1094.35
system time	13.4691
max virtual memory	3.7777756E7
max residence set size	1.5082148E7

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/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 968 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 455 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 121 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 114 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 116 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 109 ms]
            (26) typed CpxTrs


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


The TRS R consists of the following rules:

   active(zeros) -> mark(cons(0, zeros))
   active(U11(tt, L)) -> mark(s(length(L)))
   active(and(tt, X)) -> mark(X)
   active(isNat(0)) -> mark(tt)
   active(isNat(length(V1))) -> mark(isNatList(V1))
   active(isNat(s(V1))) -> mark(isNat(V1))
   active(isNatIList(V)) -> mark(isNatList(V))
   active(isNatIList(zeros)) -> mark(tt)
   active(isNatIList(cons(V1, V2))) -> mark(and(isNat(V1), isNatIList(V2)))
   active(isNatList(nil)) -> mark(tt)
   active(isNatList(cons(V1, V2))) -> mark(and(isNat(V1), isNatList(V2)))
   active(length(nil)) -> mark(0)
   active(length(cons(N, L))) -> mark(U11(and(isNatList(L), isNat(N)), L))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(U11(X1, X2)) -> U11(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(length(X)) -> length(active(X))
   active(and(X1, X2)) -> and(active(X1), X2)
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   U11(mark(X1), X2) -> mark(U11(X1, X2))
   s(mark(X)) -> mark(s(X))
   length(mark(X)) -> mark(length(X))
   and(mark(X1), X2) -> mark(and(X1, X2))
   proper(zeros) -> ok(zeros)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(U11(X1, X2)) -> U11(proper(X1), proper(X2))
   proper(tt) -> ok(tt)
   proper(s(X)) -> s(proper(X))
   proper(length(X)) -> length(proper(X))
   proper(and(X1, X2)) -> and(proper(X1), proper(X2))
   proper(isNat(X)) -> isNat(proper(X))
   proper(isNatList(X)) -> isNatList(proper(X))
   proper(isNatIList(X)) -> isNatIList(proper(X))
   proper(nil) -> ok(nil)
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   U11(ok(X1), ok(X2)) -> ok(U11(X1, X2))
   s(ok(X)) -> ok(s(X))
   length(ok(X)) -> ok(length(X))
   and(ok(X1), ok(X2)) -> ok(and(X1, X2))
   isNat(ok(X)) -> ok(isNat(X))
   isNatList(ok(X)) -> ok(isNatList(X))
   isNatIList(ok(X)) -> ok(isNatIList(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of cons: active, proper, cons
The following defined symbols can occur below the 1th argument of cons: active, proper, cons
The following defined symbols can occur below the 0th argument of top: active, proper, cons
The following defined symbols can occur below the 0th argument of proper: active, proper, cons

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