Runtime Complexity: TRS pair #487110488

details

property	value
status	complete
benchmark	Ex3_2_Luc97_C.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n148.star.cs.uiowa.edu
space	Transformed_CSR_04

run statistics

property	value
solver	AProVE
configuration	complexity
runtime (wallclock)	13.2107 seconds
cpu usage	29.2888
user time	27.4958
system time	1.79295
max virtual memory	1.9339884E7
max residence set size	4199352.0

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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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, 69 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), 445 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), 70 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 123 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 72 ms]
            (24) typed CpxTrs


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

(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(dbl(0)) -> mark(0)
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   active(dbls(nil)) -> mark(nil)
   active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
   active(sel(0, cons(X, Y))) -> mark(X)
   active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
   active(indx(nil, X)) -> mark(nil)
   active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
   active(from(X)) -> mark(cons(X, from(s(X))))
   active(dbl(X)) -> dbl(active(X))
   active(dbls(X)) -> dbls(active(X))
   active(sel(X1, X2)) -> sel(active(X1), X2)
   active(sel(X1, X2)) -> sel(X1, active(X2))
   active(indx(X1, X2)) -> indx(active(X1), X2)
   dbl(mark(X)) -> mark(dbl(X))
   dbls(mark(X)) -> mark(dbls(X))
   sel(mark(X1), X2) -> mark(sel(X1, X2))
   sel(X1, mark(X2)) -> mark(sel(X1, X2))
   indx(mark(X1), X2) -> mark(indx(X1, X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(dbls(X)) -> dbls(proper(X))
   proper(nil) -> ok(nil)
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(sel(X1, X2)) -> sel(proper(X1), proper(X2))
   proper(indx(X1, X2)) -> indx(proper(X1), proper(X2))
   proper(from(X)) -> from(proper(X))
   dbl(ok(X)) -> ok(dbl(X))
   s(ok(X)) -> ok(s(X))
   dbls(ok(X)) -> ok(dbls(X))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   sel(ok(X1), ok(X2)) -> ok(sel(X1, X2))
   indx(ok(X1), ok(X2)) -> ok(indx(X1, X2))
   from(ok(X)) -> ok(from(X))
   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(dbl(0)) -> mark(0)
active(dbl(s(X))) -> mark(s(s(dbl(X))))
active(dbls(nil)) -> mark(nil)
active(dbls(cons(X, Y))) -> mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) -> mark(X)
active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z))
active(indx(nil, X)) -> mark(nil)
active(indx(cons(X, Y), Z)) -> mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) -> mark(cons(X, from(s(X))))

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