1060.78/292.34 WORST_CASE(Omega(n^1), ?) 1061.00/292.35 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1061.00/292.35 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1061.00/292.35 1061.00/292.35 1061.00/292.35 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1061.00/292.35 1061.00/292.35 (0) CpxTRS 1061.00/292.35 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1061.00/292.35 (2) TRS for Loop Detection 1061.00/292.35 (3) DecreasingLoopProof [LOWER BOUND(ID), 33 ms] 1061.00/292.35 (4) BEST 1061.00/292.35 (5) proven lower bound 1061.00/292.35 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1061.00/292.35 (7) BOUNDS(n^1, INF) 1061.00/292.35 (8) TRS for Loop Detection 1061.00/292.35 1061.00/292.35 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (0) 1061.00/292.35 Obligation: 1061.00/292.35 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1061.00/292.35 1061.00/292.35 1061.00/292.35 The TRS R consists of the following rules: 1061.00/292.35 1061.00/292.35 filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) 1061.00/292.35 filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) 1061.00/292.35 sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) 1061.00/292.35 sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N))) 1061.00/292.35 nats(N) -> cons(N, n__nats(s(N))) 1061.00/292.35 zprimes -> sieve(nats(s(s(0)))) 1061.00/292.35 filter(X1, X2, X3) -> n__filter(X1, X2, X3) 1061.00/292.35 sieve(X) -> n__sieve(X) 1061.00/292.35 nats(X) -> n__nats(X) 1061.00/292.35 activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3) 1061.00/292.35 activate(n__sieve(X)) -> sieve(X) 1061.00/292.35 activate(n__nats(X)) -> nats(X) 1061.00/292.35 activate(X) -> X 1061.00/292.35 1061.00/292.35 S is empty. 1061.00/292.35 Rewrite Strategy: INNERMOST 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1061.00/292.35 Transformed a relative TRS into a decreasing-loop problem. 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (2) 1061.00/292.35 Obligation: 1061.00/292.35 Analyzing the following TRS for decreasing loops: 1061.00/292.35 1061.00/292.35 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1061.00/292.35 1061.00/292.35 1061.00/292.35 The TRS R consists of the following rules: 1061.00/292.35 1061.00/292.35 filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) 1061.00/292.35 filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) 1061.00/292.35 sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) 1061.00/292.35 sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N))) 1061.00/292.35 nats(N) -> cons(N, n__nats(s(N))) 1061.00/292.35 zprimes -> sieve(nats(s(s(0)))) 1061.00/292.35 filter(X1, X2, X3) -> n__filter(X1, X2, X3) 1061.00/292.35 sieve(X) -> n__sieve(X) 1061.00/292.35 nats(X) -> n__nats(X) 1061.00/292.35 activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3) 1061.00/292.35 activate(n__sieve(X)) -> sieve(X) 1061.00/292.35 activate(n__nats(X)) -> nats(X) 1061.00/292.35 activate(X) -> X 1061.00/292.35 1061.00/292.35 S is empty. 1061.00/292.35 Rewrite Strategy: INNERMOST 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1061.00/292.35 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1061.00/292.35 1061.00/292.35 The rewrite sequence 1061.00/292.35 1061.00/292.35 sieve(cons(s(N), n__sieve(X1_0))) ->^+ cons(s(N), n__sieve(filter(sieve(X1_0), N, N))) 1061.00/292.35 1061.00/292.35 gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0]. 1061.00/292.35 1061.00/292.35 The pumping substitution is [X1_0 / cons(s(N), n__sieve(X1_0))]. 1061.00/292.35 1061.00/292.35 The result substitution is [ ]. 1061.00/292.35 1061.00/292.35 1061.00/292.35 1061.00/292.35 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (4) 1061.00/292.35 Complex Obligation (BEST) 1061.00/292.35 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (5) 1061.00/292.35 Obligation: 1061.00/292.35 Proved the lower bound n^1 for the following obligation: 1061.00/292.35 1061.00/292.35 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1061.00/292.35 1061.00/292.35 1061.00/292.35 The TRS R consists of the following rules: 1061.00/292.35 1061.00/292.35 filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) 1061.00/292.35 filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) 1061.00/292.35 sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) 1061.00/292.35 sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N))) 1061.00/292.35 nats(N) -> cons(N, n__nats(s(N))) 1061.00/292.35 zprimes -> sieve(nats(s(s(0)))) 1061.00/292.35 filter(X1, X2, X3) -> n__filter(X1, X2, X3) 1061.00/292.35 sieve(X) -> n__sieve(X) 1061.00/292.35 nats(X) -> n__nats(X) 1061.00/292.35 activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3) 1061.00/292.35 activate(n__sieve(X)) -> sieve(X) 1061.00/292.35 activate(n__nats(X)) -> nats(X) 1061.00/292.35 activate(X) -> X 1061.00/292.35 1061.00/292.35 S is empty. 1061.00/292.35 Rewrite Strategy: INNERMOST 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (6) LowerBoundPropagationProof (FINISHED) 1061.00/292.35 Propagated lower bound. 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (7) 1061.00/292.35 BOUNDS(n^1, INF) 1061.00/292.35 1061.00/292.35 ---------------------------------------- 1061.00/292.35 1061.00/292.35 (8) 1061.00/292.35 Obligation: 1061.00/292.35 Analyzing the following TRS for decreasing loops: 1061.00/292.35 1061.00/292.35 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1061.00/292.35 1061.00/292.35 1061.00/292.35 The TRS R consists of the following rules: 1061.00/292.35 1061.00/292.35 filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M)) 1061.00/292.35 filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M)) 1061.00/292.35 sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y))) 1061.00/292.35 sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N))) 1061.00/292.35 nats(N) -> cons(N, n__nats(s(N))) 1061.00/292.35 zprimes -> sieve(nats(s(s(0)))) 1061.00/292.35 filter(X1, X2, X3) -> n__filter(X1, X2, X3) 1061.00/292.35 sieve(X) -> n__sieve(X) 1061.00/292.35 nats(X) -> n__nats(X) 1061.00/292.35 activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3) 1061.00/292.35 activate(n__sieve(X)) -> sieve(X) 1061.00/292.35 activate(n__nats(X)) -> nats(X) 1061.00/292.35 activate(X) -> X 1061.00/292.35 1061.00/292.35 S is empty. 1061.00/292.35 Rewrite Strategy: INNERMOST 1061.16/292.44 EOF