Runtime Complexity: TRS pair #487110392

details

property	value
status	complete
benchmark	Ex8_BLR02_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)	8.02704 seconds
cpu usage	23.7337
user time	22.144
system time	1.58969
max virtual memory	1.9144812E7
max residence set size	3772696.0

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/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, 114 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) TRS for Loop Detection


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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(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
   active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
   active(add(0, X)) -> mark(X)
   active(add(s(X), Y)) -> mark(s(add(X, Y)))
   active(sel(0, cons(X, XS))) -> mark(X)
   active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
   active(fib(X)) -> fib(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(fib1(X1, X2)) -> fib1(active(X1), X2)
   active(fib1(X1, X2)) -> fib1(X1, active(X2))
   active(s(X)) -> s(active(X))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(add(X1, X2)) -> add(active(X1), X2)
   active(add(X1, X2)) -> add(X1, active(X2))
   fib(mark(X)) -> mark(fib(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   fib1(mark(X1), X2) -> mark(fib1(X1, X2))
   fib1(X1, mark(X2)) -> mark(fib1(X1, X2))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   add(mark(X1), X2) -> mark(add(X1, X2))
   add(X1, mark(X2)) -> mark(add(X1, X2))
   proper(fib(X)) -> fib(proper(X))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(fib1(X1, X2)) -> fib1(proper(X1), proper(X2))
   proper(s(X)) -> s(proper(X))
   proper(0) -> ok(0)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(add(X1, X2)) -> add(proper(X1), proper(X2))
   fib(ok(X)) -> ok(fib(X))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   fib1(ok(X1), ok(X2)) -> ok(fib1(X1, X2))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   add(ok(X1), ok(X2)) -> ok(add(X1, X2))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
active(fib(N)) -> mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) -> mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) -> mark(X)
active(add(s(X), Y)) -> mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) -> mark(X)
active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS))
active(fib(X)) -> fib(active(X))
active(sel(X1, X2)) -> sel(active(X1), X2)
active(sel(X1, X2)) -> sel(X1, active(X2))
active(fib1(X1, X2)) -> fib1(active(X1), X2)
active(fib1(X1, X2)) -> fib1(X1, active(X2))
active(s(X)) -> s(active(X))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(add(X1, X2)) -> add(active(X1), X2)
active(add(X1, X2)) -> add(X1, active(X2))
proper(fib(X)) -> fib(proper(X))
proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))

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