1117.37/291.51 WORST_CASE(Omega(n^1), ?) 1137.59/296.58 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1137.59/296.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1137.59/296.58 1137.59/296.58 1137.59/296.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1137.59/296.58 1137.59/296.58 (0) CpxTRS 1137.59/296.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1137.59/296.58 (2) TRS for Loop Detection 1137.59/296.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1137.59/296.58 (4) BEST 1137.59/296.58 (5) proven lower bound 1137.59/296.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1137.59/296.58 (7) BOUNDS(n^1, INF) 1137.59/296.58 (8) TRS for Loop Detection 1137.59/296.58 1137.59/296.58 1137.59/296.58 ---------------------------------------- 1137.59/296.58 1137.59/296.58 (0) 1137.59/296.58 Obligation: 1137.59/296.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1137.59/296.58 1137.59/296.58 1137.59/296.58 The TRS R consists of the following rules: 1137.59/296.58 1137.59/296.58 a__from(X) -> cons(mark(X), from(s(X))) 1137.59/296.58 a__after(0, XS) -> mark(XS) 1137.59/296.58 a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS)) 1137.59/296.58 mark(from(X)) -> a__from(mark(X)) 1137.59/296.58 mark(after(X1, X2)) -> a__after(mark(X1), mark(X2)) 1137.59/296.58 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1137.59/296.59 mark(s(X)) -> s(mark(X)) 1137.59/296.59 mark(0) -> 0 1137.59/296.59 a__from(X) -> from(X) 1137.59/296.59 a__after(X1, X2) -> after(X1, X2) 1137.59/296.59 1137.59/296.59 S is empty. 1137.59/296.59 Rewrite Strategy: INNERMOST 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1137.59/296.59 Transformed a relative TRS into a decreasing-loop problem. 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (2) 1137.59/296.59 Obligation: 1137.59/296.59 Analyzing the following TRS for decreasing loops: 1137.59/296.59 1137.59/296.59 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1137.59/296.59 1137.59/296.59 1137.59/296.59 The TRS R consists of the following rules: 1137.59/296.59 1137.59/296.59 a__from(X) -> cons(mark(X), from(s(X))) 1137.59/296.59 a__after(0, XS) -> mark(XS) 1137.59/296.59 a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS)) 1137.59/296.59 mark(from(X)) -> a__from(mark(X)) 1137.59/296.59 mark(after(X1, X2)) -> a__after(mark(X1), mark(X2)) 1137.59/296.59 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1137.59/296.59 mark(s(X)) -> s(mark(X)) 1137.59/296.59 mark(0) -> 0 1137.59/296.59 a__from(X) -> from(X) 1137.59/296.59 a__after(X1, X2) -> after(X1, X2) 1137.59/296.59 1137.59/296.59 S is empty. 1137.59/296.59 Rewrite Strategy: INNERMOST 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1137.59/296.59 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1137.59/296.59 1137.59/296.59 The rewrite sequence 1137.59/296.59 1137.59/296.59 mark(from(X)) ->^+ a__from(mark(X)) 1137.59/296.59 1137.59/296.59 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1137.59/296.59 1137.59/296.59 The pumping substitution is [X / from(X)]. 1137.59/296.59 1137.59/296.59 The result substitution is [ ]. 1137.59/296.59 1137.59/296.59 1137.59/296.59 1137.59/296.59 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (4) 1137.59/296.59 Complex Obligation (BEST) 1137.59/296.59 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (5) 1137.59/296.59 Obligation: 1137.59/296.59 Proved the lower bound n^1 for the following obligation: 1137.59/296.59 1137.59/296.59 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1137.59/296.59 1137.59/296.59 1137.59/296.59 The TRS R consists of the following rules: 1137.59/296.59 1137.59/296.59 a__from(X) -> cons(mark(X), from(s(X))) 1137.59/296.59 a__after(0, XS) -> mark(XS) 1137.59/296.59 a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS)) 1137.59/296.59 mark(from(X)) -> a__from(mark(X)) 1137.59/296.59 mark(after(X1, X2)) -> a__after(mark(X1), mark(X2)) 1137.59/296.59 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1137.59/296.59 mark(s(X)) -> s(mark(X)) 1137.59/296.59 mark(0) -> 0 1137.59/296.59 a__from(X) -> from(X) 1137.59/296.59 a__after(X1, X2) -> after(X1, X2) 1137.59/296.59 1137.59/296.59 S is empty. 1137.59/296.59 Rewrite Strategy: INNERMOST 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (6) LowerBoundPropagationProof (FINISHED) 1137.59/296.59 Propagated lower bound. 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (7) 1137.59/296.59 BOUNDS(n^1, INF) 1137.59/296.59 1137.59/296.59 ---------------------------------------- 1137.59/296.59 1137.59/296.59 (8) 1137.59/296.59 Obligation: 1137.59/296.59 Analyzing the following TRS for decreasing loops: 1137.59/296.59 1137.59/296.59 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1137.59/296.59 1137.59/296.59 1137.59/296.59 The TRS R consists of the following rules: 1137.59/296.59 1137.59/296.59 a__from(X) -> cons(mark(X), from(s(X))) 1137.59/296.59 a__after(0, XS) -> mark(XS) 1137.59/296.59 a__after(s(N), cons(X, XS)) -> a__after(mark(N), mark(XS)) 1137.59/296.59 mark(from(X)) -> a__from(mark(X)) 1137.59/296.59 mark(after(X1, X2)) -> a__after(mark(X1), mark(X2)) 1137.59/296.59 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1137.59/296.59 mark(s(X)) -> s(mark(X)) 1137.59/296.59 mark(0) -> 0 1137.59/296.59 a__from(X) -> from(X) 1137.59/296.59 a__after(X1, X2) -> after(X1, X2) 1137.59/296.59 1137.59/296.59 S is empty. 1137.59/296.59 Rewrite Strategy: INNERMOST 1137.68/296.65 EOF