1126.45/291.58 WORST_CASE(Omega(n^1), ?) 1128.44/292.07 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1128.44/292.07 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1128.44/292.07 1128.44/292.07 1128.44/292.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1128.44/292.07 1128.44/292.07 (0) CpxTRS 1128.44/292.07 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1128.44/292.07 (2) TRS for Loop Detection 1128.44/292.07 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1128.44/292.07 (4) BEST 1128.44/292.07 (5) proven lower bound 1128.44/292.07 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1128.44/292.07 (7) BOUNDS(n^1, INF) 1128.44/292.07 (8) TRS for Loop Detection 1128.44/292.07 1128.44/292.07 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (0) 1128.44/292.07 Obligation: 1128.44/292.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1128.44/292.07 1128.44/292.07 1128.44/292.07 The TRS R consists of the following rules: 1128.44/292.07 1128.44/292.07 active(pairNs) -> mark(cons(0, incr(oddNs))) 1128.44/292.07 active(oddNs) -> mark(incr(pairNs)) 1128.44/292.07 active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS))) 1128.44/292.07 active(take(0, XS)) -> mark(nil) 1128.44/292.07 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 1128.44/292.07 active(zip(nil, XS)) -> mark(nil) 1128.44/292.07 active(zip(X, nil)) -> mark(nil) 1128.44/292.07 active(zip(cons(X, XS), cons(Y, YS))) -> mark(cons(pair(X, Y), zip(XS, YS))) 1128.44/292.07 active(tail(cons(X, XS))) -> mark(XS) 1128.44/292.07 active(repItems(nil)) -> mark(nil) 1128.44/292.07 active(repItems(cons(X, XS))) -> mark(cons(X, cons(X, repItems(XS)))) 1128.44/292.07 active(cons(X1, X2)) -> cons(active(X1), X2) 1128.44/292.07 active(incr(X)) -> incr(active(X)) 1128.44/292.07 active(s(X)) -> s(active(X)) 1128.44/292.07 active(take(X1, X2)) -> take(active(X1), X2) 1128.44/292.07 active(take(X1, X2)) -> take(X1, active(X2)) 1128.44/292.07 active(zip(X1, X2)) -> zip(active(X1), X2) 1128.44/292.07 active(zip(X1, X2)) -> zip(X1, active(X2)) 1128.44/292.07 active(pair(X1, X2)) -> pair(active(X1), X2) 1128.44/292.07 active(pair(X1, X2)) -> pair(X1, active(X2)) 1128.44/292.07 active(tail(X)) -> tail(active(X)) 1128.44/292.07 active(repItems(X)) -> repItems(active(X)) 1128.44/292.07 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1128.44/292.07 incr(mark(X)) -> mark(incr(X)) 1128.44/292.07 s(mark(X)) -> mark(s(X)) 1128.44/292.07 take(mark(X1), X2) -> mark(take(X1, X2)) 1128.44/292.07 take(X1, mark(X2)) -> mark(take(X1, X2)) 1128.44/292.07 zip(mark(X1), X2) -> mark(zip(X1, X2)) 1128.44/292.07 zip(X1, mark(X2)) -> mark(zip(X1, X2)) 1128.44/292.07 pair(mark(X1), X2) -> mark(pair(X1, X2)) 1128.44/292.07 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 1128.44/292.07 tail(mark(X)) -> mark(tail(X)) 1128.44/292.07 repItems(mark(X)) -> mark(repItems(X)) 1128.44/292.07 proper(pairNs) -> ok(pairNs) 1128.44/292.07 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1128.44/292.07 proper(0) -> ok(0) 1128.44/292.07 proper(incr(X)) -> incr(proper(X)) 1128.44/292.07 proper(oddNs) -> ok(oddNs) 1128.44/292.07 proper(s(X)) -> s(proper(X)) 1128.44/292.07 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1128.44/292.07 proper(nil) -> ok(nil) 1128.44/292.07 proper(zip(X1, X2)) -> zip(proper(X1), proper(X2)) 1128.44/292.07 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 1128.44/292.07 proper(tail(X)) -> tail(proper(X)) 1128.44/292.07 proper(repItems(X)) -> repItems(proper(X)) 1128.44/292.07 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1128.44/292.07 incr(ok(X)) -> ok(incr(X)) 1128.44/292.07 s(ok(X)) -> ok(s(X)) 1128.44/292.07 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1128.44/292.07 zip(ok(X1), ok(X2)) -> ok(zip(X1, X2)) 1128.44/292.07 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 1128.44/292.07 tail(ok(X)) -> ok(tail(X)) 1128.44/292.07 repItems(ok(X)) -> ok(repItems(X)) 1128.44/292.07 top(mark(X)) -> top(proper(X)) 1128.44/292.07 top(ok(X)) -> top(active(X)) 1128.44/292.07 1128.44/292.07 S is empty. 1128.44/292.07 Rewrite Strategy: FULL 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1128.44/292.07 Transformed a relative TRS into a decreasing-loop problem. 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (2) 1128.44/292.07 Obligation: 1128.44/292.07 Analyzing the following TRS for decreasing loops: 1128.44/292.07 1128.44/292.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1128.44/292.07 1128.44/292.07 1128.44/292.07 The TRS R consists of the following rules: 1128.44/292.07 1128.44/292.07 active(pairNs) -> mark(cons(0, incr(oddNs))) 1128.44/292.07 active(oddNs) -> mark(incr(pairNs)) 1128.44/292.07 active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS))) 1128.44/292.07 active(take(0, XS)) -> mark(nil) 1128.44/292.07 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 1128.44/292.07 active(zip(nil, XS)) -> mark(nil) 1128.44/292.07 active(zip(X, nil)) -> mark(nil) 1128.44/292.07 active(zip(cons(X, XS), cons(Y, YS))) -> mark(cons(pair(X, Y), zip(XS, YS))) 1128.44/292.07 active(tail(cons(X, XS))) -> mark(XS) 1128.44/292.07 active(repItems(nil)) -> mark(nil) 1128.44/292.07 active(repItems(cons(X, XS))) -> mark(cons(X, cons(X, repItems(XS)))) 1128.44/292.07 active(cons(X1, X2)) -> cons(active(X1), X2) 1128.44/292.07 active(incr(X)) -> incr(active(X)) 1128.44/292.07 active(s(X)) -> s(active(X)) 1128.44/292.07 active(take(X1, X2)) -> take(active(X1), X2) 1128.44/292.07 active(take(X1, X2)) -> take(X1, active(X2)) 1128.44/292.07 active(zip(X1, X2)) -> zip(active(X1), X2) 1128.44/292.07 active(zip(X1, X2)) -> zip(X1, active(X2)) 1128.44/292.07 active(pair(X1, X2)) -> pair(active(X1), X2) 1128.44/292.07 active(pair(X1, X2)) -> pair(X1, active(X2)) 1128.44/292.07 active(tail(X)) -> tail(active(X)) 1128.44/292.07 active(repItems(X)) -> repItems(active(X)) 1128.44/292.07 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1128.44/292.07 incr(mark(X)) -> mark(incr(X)) 1128.44/292.07 s(mark(X)) -> mark(s(X)) 1128.44/292.07 take(mark(X1), X2) -> mark(take(X1, X2)) 1128.44/292.07 take(X1, mark(X2)) -> mark(take(X1, X2)) 1128.44/292.07 zip(mark(X1), X2) -> mark(zip(X1, X2)) 1128.44/292.07 zip(X1, mark(X2)) -> mark(zip(X1, X2)) 1128.44/292.07 pair(mark(X1), X2) -> mark(pair(X1, X2)) 1128.44/292.07 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 1128.44/292.07 tail(mark(X)) -> mark(tail(X)) 1128.44/292.07 repItems(mark(X)) -> mark(repItems(X)) 1128.44/292.07 proper(pairNs) -> ok(pairNs) 1128.44/292.07 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1128.44/292.07 proper(0) -> ok(0) 1128.44/292.07 proper(incr(X)) -> incr(proper(X)) 1128.44/292.07 proper(oddNs) -> ok(oddNs) 1128.44/292.07 proper(s(X)) -> s(proper(X)) 1128.44/292.07 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1128.44/292.07 proper(nil) -> ok(nil) 1128.44/292.07 proper(zip(X1, X2)) -> zip(proper(X1), proper(X2)) 1128.44/292.07 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 1128.44/292.07 proper(tail(X)) -> tail(proper(X)) 1128.44/292.07 proper(repItems(X)) -> repItems(proper(X)) 1128.44/292.07 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1128.44/292.07 incr(ok(X)) -> ok(incr(X)) 1128.44/292.07 s(ok(X)) -> ok(s(X)) 1128.44/292.07 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1128.44/292.07 zip(ok(X1), ok(X2)) -> ok(zip(X1, X2)) 1128.44/292.07 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 1128.44/292.07 tail(ok(X)) -> ok(tail(X)) 1128.44/292.07 repItems(ok(X)) -> ok(repItems(X)) 1128.44/292.07 top(mark(X)) -> top(proper(X)) 1128.44/292.07 top(ok(X)) -> top(active(X)) 1128.44/292.07 1128.44/292.07 S is empty. 1128.44/292.07 Rewrite Strategy: FULL 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1128.44/292.07 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1128.44/292.07 1128.44/292.07 The rewrite sequence 1128.44/292.07 1128.44/292.07 take(ok(X1), ok(X2)) ->^+ ok(take(X1, X2)) 1128.44/292.07 1128.44/292.07 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1128.44/292.07 1128.44/292.07 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 1128.44/292.07 1128.44/292.07 The result substitution is [ ]. 1128.44/292.07 1128.44/292.07 1128.44/292.07 1128.44/292.07 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (4) 1128.44/292.07 Complex Obligation (BEST) 1128.44/292.07 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (5) 1128.44/292.07 Obligation: 1128.44/292.07 Proved the lower bound n^1 for the following obligation: 1128.44/292.07 1128.44/292.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1128.44/292.07 1128.44/292.07 1128.44/292.07 The TRS R consists of the following rules: 1128.44/292.07 1128.44/292.07 active(pairNs) -> mark(cons(0, incr(oddNs))) 1128.44/292.07 active(oddNs) -> mark(incr(pairNs)) 1128.44/292.07 active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS))) 1128.44/292.07 active(take(0, XS)) -> mark(nil) 1128.44/292.07 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 1128.44/292.07 active(zip(nil, XS)) -> mark(nil) 1128.44/292.07 active(zip(X, nil)) -> mark(nil) 1128.44/292.07 active(zip(cons(X, XS), cons(Y, YS))) -> mark(cons(pair(X, Y), zip(XS, YS))) 1128.44/292.07 active(tail(cons(X, XS))) -> mark(XS) 1128.44/292.07 active(repItems(nil)) -> mark(nil) 1128.44/292.07 active(repItems(cons(X, XS))) -> mark(cons(X, cons(X, repItems(XS)))) 1128.44/292.07 active(cons(X1, X2)) -> cons(active(X1), X2) 1128.44/292.07 active(incr(X)) -> incr(active(X)) 1128.44/292.07 active(s(X)) -> s(active(X)) 1128.44/292.07 active(take(X1, X2)) -> take(active(X1), X2) 1128.44/292.07 active(take(X1, X2)) -> take(X1, active(X2)) 1128.44/292.07 active(zip(X1, X2)) -> zip(active(X1), X2) 1128.44/292.07 active(zip(X1, X2)) -> zip(X1, active(X2)) 1128.44/292.07 active(pair(X1, X2)) -> pair(active(X1), X2) 1128.44/292.07 active(pair(X1, X2)) -> pair(X1, active(X2)) 1128.44/292.07 active(tail(X)) -> tail(active(X)) 1128.44/292.07 active(repItems(X)) -> repItems(active(X)) 1128.44/292.07 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1128.44/292.07 incr(mark(X)) -> mark(incr(X)) 1128.44/292.07 s(mark(X)) -> mark(s(X)) 1128.44/292.07 take(mark(X1), X2) -> mark(take(X1, X2)) 1128.44/292.07 take(X1, mark(X2)) -> mark(take(X1, X2)) 1128.44/292.07 zip(mark(X1), X2) -> mark(zip(X1, X2)) 1128.44/292.07 zip(X1, mark(X2)) -> mark(zip(X1, X2)) 1128.44/292.07 pair(mark(X1), X2) -> mark(pair(X1, X2)) 1128.44/292.07 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 1128.44/292.07 tail(mark(X)) -> mark(tail(X)) 1128.44/292.07 repItems(mark(X)) -> mark(repItems(X)) 1128.44/292.07 proper(pairNs) -> ok(pairNs) 1128.44/292.07 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1128.44/292.07 proper(0) -> ok(0) 1128.44/292.07 proper(incr(X)) -> incr(proper(X)) 1128.44/292.07 proper(oddNs) -> ok(oddNs) 1128.44/292.07 proper(s(X)) -> s(proper(X)) 1128.44/292.07 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1128.44/292.07 proper(nil) -> ok(nil) 1128.44/292.07 proper(zip(X1, X2)) -> zip(proper(X1), proper(X2)) 1128.44/292.07 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 1128.44/292.07 proper(tail(X)) -> tail(proper(X)) 1128.44/292.07 proper(repItems(X)) -> repItems(proper(X)) 1128.44/292.07 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1128.44/292.07 incr(ok(X)) -> ok(incr(X)) 1128.44/292.07 s(ok(X)) -> ok(s(X)) 1128.44/292.07 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1128.44/292.07 zip(ok(X1), ok(X2)) -> ok(zip(X1, X2)) 1128.44/292.07 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 1128.44/292.07 tail(ok(X)) -> ok(tail(X)) 1128.44/292.07 repItems(ok(X)) -> ok(repItems(X)) 1128.44/292.07 top(mark(X)) -> top(proper(X)) 1128.44/292.07 top(ok(X)) -> top(active(X)) 1128.44/292.07 1128.44/292.07 S is empty. 1128.44/292.07 Rewrite Strategy: FULL 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (6) LowerBoundPropagationProof (FINISHED) 1128.44/292.07 Propagated lower bound. 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (7) 1128.44/292.07 BOUNDS(n^1, INF) 1128.44/292.07 1128.44/292.07 ---------------------------------------- 1128.44/292.07 1128.44/292.07 (8) 1128.44/292.07 Obligation: 1128.44/292.07 Analyzing the following TRS for decreasing loops: 1128.44/292.07 1128.44/292.07 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1128.44/292.07 1128.44/292.07 1128.44/292.07 The TRS R consists of the following rules: 1128.44/292.07 1128.44/292.07 active(pairNs) -> mark(cons(0, incr(oddNs))) 1128.44/292.07 active(oddNs) -> mark(incr(pairNs)) 1128.44/292.07 active(incr(cons(X, XS))) -> mark(cons(s(X), incr(XS))) 1128.44/292.07 active(take(0, XS)) -> mark(nil) 1128.44/292.07 active(take(s(N), cons(X, XS))) -> mark(cons(X, take(N, XS))) 1128.44/292.07 active(zip(nil, XS)) -> mark(nil) 1128.44/292.07 active(zip(X, nil)) -> mark(nil) 1128.44/292.07 active(zip(cons(X, XS), cons(Y, YS))) -> mark(cons(pair(X, Y), zip(XS, YS))) 1128.44/292.07 active(tail(cons(X, XS))) -> mark(XS) 1128.44/292.07 active(repItems(nil)) -> mark(nil) 1128.44/292.07 active(repItems(cons(X, XS))) -> mark(cons(X, cons(X, repItems(XS)))) 1128.44/292.07 active(cons(X1, X2)) -> cons(active(X1), X2) 1128.44/292.07 active(incr(X)) -> incr(active(X)) 1128.44/292.07 active(s(X)) -> s(active(X)) 1128.44/292.07 active(take(X1, X2)) -> take(active(X1), X2) 1128.44/292.07 active(take(X1, X2)) -> take(X1, active(X2)) 1128.44/292.07 active(zip(X1, X2)) -> zip(active(X1), X2) 1128.44/292.07 active(zip(X1, X2)) -> zip(X1, active(X2)) 1128.44/292.07 active(pair(X1, X2)) -> pair(active(X1), X2) 1128.44/292.07 active(pair(X1, X2)) -> pair(X1, active(X2)) 1128.44/292.07 active(tail(X)) -> tail(active(X)) 1128.44/292.07 active(repItems(X)) -> repItems(active(X)) 1128.44/292.07 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1128.44/292.07 incr(mark(X)) -> mark(incr(X)) 1128.44/292.07 s(mark(X)) -> mark(s(X)) 1128.44/292.07 take(mark(X1), X2) -> mark(take(X1, X2)) 1128.44/292.07 take(X1, mark(X2)) -> mark(take(X1, X2)) 1128.44/292.07 zip(mark(X1), X2) -> mark(zip(X1, X2)) 1128.44/292.07 zip(X1, mark(X2)) -> mark(zip(X1, X2)) 1128.44/292.07 pair(mark(X1), X2) -> mark(pair(X1, X2)) 1128.44/292.07 pair(X1, mark(X2)) -> mark(pair(X1, X2)) 1128.44/292.07 tail(mark(X)) -> mark(tail(X)) 1128.44/292.07 repItems(mark(X)) -> mark(repItems(X)) 1128.44/292.07 proper(pairNs) -> ok(pairNs) 1128.44/292.07 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1128.44/292.07 proper(0) -> ok(0) 1128.44/292.07 proper(incr(X)) -> incr(proper(X)) 1128.44/292.07 proper(oddNs) -> ok(oddNs) 1128.44/292.07 proper(s(X)) -> s(proper(X)) 1128.44/292.07 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1128.44/292.07 proper(nil) -> ok(nil) 1128.44/292.07 proper(zip(X1, X2)) -> zip(proper(X1), proper(X2)) 1128.44/292.07 proper(pair(X1, X2)) -> pair(proper(X1), proper(X2)) 1128.44/292.07 proper(tail(X)) -> tail(proper(X)) 1128.44/292.07 proper(repItems(X)) -> repItems(proper(X)) 1128.44/292.07 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1128.44/292.07 incr(ok(X)) -> ok(incr(X)) 1128.44/292.07 s(ok(X)) -> ok(s(X)) 1128.44/292.07 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1128.44/292.07 zip(ok(X1), ok(X2)) -> ok(zip(X1, X2)) 1128.44/292.07 pair(ok(X1), ok(X2)) -> ok(pair(X1, X2)) 1128.44/292.07 tail(ok(X)) -> ok(tail(X)) 1128.44/292.07 repItems(ok(X)) -> ok(repItems(X)) 1128.44/292.07 top(mark(X)) -> top(proper(X)) 1128.44/292.07 top(ok(X)) -> top(active(X)) 1128.44/292.07 1128.44/292.07 S is empty. 1128.44/292.07 Rewrite Strategy: FULL 1128.59/292.13 EOF