1155.12/299.98 WORST_CASE(Omega(n^1), ?) 1155.12/300.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1155.12/300.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1155.12/300.02 1155.12/300.02 1155.12/300.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1155.12/300.02 1155.12/300.02 (0) CpxTRS 1155.12/300.02 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1155.12/300.02 (2) TRS for Loop Detection 1155.12/300.02 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1155.12/300.02 (4) BEST 1155.12/300.02 (5) proven lower bound 1155.12/300.02 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1155.12/300.02 (7) BOUNDS(n^1, INF) 1155.12/300.02 (8) TRS for Loop Detection 1155.12/300.02 1155.12/300.02 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (0) 1155.12/300.02 Obligation: 1155.12/300.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1155.12/300.02 1155.12/300.02 1155.12/300.02 The TRS R consists of the following rules: 1155.12/300.02 1155.12/300.02 a__from(X) -> cons(mark(X), from(s(X))) 1155.12/300.02 a__head(cons(X, XS)) -> mark(X) 1155.12/300.02 a__2nd(cons(X, XS)) -> a__head(mark(XS)) 1155.12/300.02 a__take(0, XS) -> nil 1155.12/300.02 a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) 1155.12/300.02 a__sel(0, cons(X, XS)) -> mark(X) 1155.12/300.02 a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) 1155.12/300.02 mark(from(X)) -> a__from(mark(X)) 1155.12/300.02 mark(head(X)) -> a__head(mark(X)) 1155.12/300.02 mark(2nd(X)) -> a__2nd(mark(X)) 1155.12/300.02 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1155.12/300.02 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1155.12/300.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1155.12/300.02 mark(s(X)) -> s(mark(X)) 1155.12/300.02 mark(0) -> 0 1155.12/300.02 mark(nil) -> nil 1155.12/300.02 a__from(X) -> from(X) 1155.12/300.02 a__head(X) -> head(X) 1155.12/300.02 a__2nd(X) -> 2nd(X) 1155.12/300.02 a__take(X1, X2) -> take(X1, X2) 1155.12/300.02 a__sel(X1, X2) -> sel(X1, X2) 1155.12/300.02 1155.12/300.02 S is empty. 1155.12/300.02 Rewrite Strategy: INNERMOST 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1155.12/300.02 Transformed a relative TRS into a decreasing-loop problem. 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (2) 1155.12/300.02 Obligation: 1155.12/300.02 Analyzing the following TRS for decreasing loops: 1155.12/300.02 1155.12/300.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1155.12/300.02 1155.12/300.02 1155.12/300.02 The TRS R consists of the following rules: 1155.12/300.02 1155.12/300.02 a__from(X) -> cons(mark(X), from(s(X))) 1155.12/300.02 a__head(cons(X, XS)) -> mark(X) 1155.12/300.02 a__2nd(cons(X, XS)) -> a__head(mark(XS)) 1155.12/300.02 a__take(0, XS) -> nil 1155.12/300.02 a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) 1155.12/300.02 a__sel(0, cons(X, XS)) -> mark(X) 1155.12/300.02 a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) 1155.12/300.02 mark(from(X)) -> a__from(mark(X)) 1155.12/300.02 mark(head(X)) -> a__head(mark(X)) 1155.12/300.02 mark(2nd(X)) -> a__2nd(mark(X)) 1155.12/300.02 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1155.12/300.02 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1155.12/300.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1155.12/300.02 mark(s(X)) -> s(mark(X)) 1155.12/300.02 mark(0) -> 0 1155.12/300.02 mark(nil) -> nil 1155.12/300.02 a__from(X) -> from(X) 1155.12/300.02 a__head(X) -> head(X) 1155.12/300.02 a__2nd(X) -> 2nd(X) 1155.12/300.02 a__take(X1, X2) -> take(X1, X2) 1155.12/300.02 a__sel(X1, X2) -> sel(X1, X2) 1155.12/300.02 1155.12/300.02 S is empty. 1155.12/300.02 Rewrite Strategy: INNERMOST 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1155.12/300.02 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1155.12/300.02 1155.12/300.02 The rewrite sequence 1155.12/300.02 1155.12/300.02 mark(from(X)) ->^+ a__from(mark(X)) 1155.12/300.02 1155.12/300.02 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1155.12/300.02 1155.12/300.02 The pumping substitution is [X / from(X)]. 1155.12/300.02 1155.12/300.02 The result substitution is [ ]. 1155.12/300.02 1155.12/300.02 1155.12/300.02 1155.12/300.02 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (4) 1155.12/300.02 Complex Obligation (BEST) 1155.12/300.02 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (5) 1155.12/300.02 Obligation: 1155.12/300.02 Proved the lower bound n^1 for the following obligation: 1155.12/300.02 1155.12/300.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1155.12/300.02 1155.12/300.02 1155.12/300.02 The TRS R consists of the following rules: 1155.12/300.02 1155.12/300.02 a__from(X) -> cons(mark(X), from(s(X))) 1155.12/300.02 a__head(cons(X, XS)) -> mark(X) 1155.12/300.02 a__2nd(cons(X, XS)) -> a__head(mark(XS)) 1155.12/300.02 a__take(0, XS) -> nil 1155.12/300.02 a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) 1155.12/300.02 a__sel(0, cons(X, XS)) -> mark(X) 1155.12/300.02 a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) 1155.12/300.02 mark(from(X)) -> a__from(mark(X)) 1155.12/300.02 mark(head(X)) -> a__head(mark(X)) 1155.12/300.02 mark(2nd(X)) -> a__2nd(mark(X)) 1155.12/300.02 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1155.12/300.02 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1155.12/300.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1155.12/300.02 mark(s(X)) -> s(mark(X)) 1155.12/300.02 mark(0) -> 0 1155.12/300.02 mark(nil) -> nil 1155.12/300.02 a__from(X) -> from(X) 1155.12/300.02 a__head(X) -> head(X) 1155.12/300.02 a__2nd(X) -> 2nd(X) 1155.12/300.02 a__take(X1, X2) -> take(X1, X2) 1155.12/300.02 a__sel(X1, X2) -> sel(X1, X2) 1155.12/300.02 1155.12/300.02 S is empty. 1155.12/300.02 Rewrite Strategy: INNERMOST 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (6) LowerBoundPropagationProof (FINISHED) 1155.12/300.02 Propagated lower bound. 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (7) 1155.12/300.02 BOUNDS(n^1, INF) 1155.12/300.02 1155.12/300.02 ---------------------------------------- 1155.12/300.02 1155.12/300.02 (8) 1155.12/300.02 Obligation: 1155.12/300.02 Analyzing the following TRS for decreasing loops: 1155.12/300.02 1155.12/300.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1155.12/300.02 1155.12/300.02 1155.12/300.02 The TRS R consists of the following rules: 1155.12/300.02 1155.12/300.02 a__from(X) -> cons(mark(X), from(s(X))) 1155.12/300.02 a__head(cons(X, XS)) -> mark(X) 1155.12/300.02 a__2nd(cons(X, XS)) -> a__head(mark(XS)) 1155.12/300.02 a__take(0, XS) -> nil 1155.12/300.02 a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS)) 1155.12/300.02 a__sel(0, cons(X, XS)) -> mark(X) 1155.12/300.02 a__sel(s(N), cons(X, XS)) -> a__sel(mark(N), mark(XS)) 1155.12/300.02 mark(from(X)) -> a__from(mark(X)) 1155.12/300.02 mark(head(X)) -> a__head(mark(X)) 1155.12/300.02 mark(2nd(X)) -> a__2nd(mark(X)) 1155.12/300.02 mark(take(X1, X2)) -> a__take(mark(X1), mark(X2)) 1155.12/300.02 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1155.12/300.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1155.12/300.02 mark(s(X)) -> s(mark(X)) 1155.12/300.02 mark(0) -> 0 1155.12/300.02 mark(nil) -> nil 1155.12/300.02 a__from(X) -> from(X) 1155.12/300.02 a__head(X) -> head(X) 1155.12/300.02 a__2nd(X) -> 2nd(X) 1155.12/300.02 a__take(X1, X2) -> take(X1, X2) 1155.12/300.02 a__sel(X1, X2) -> sel(X1, X2) 1155.12/300.02 1155.12/300.02 S is empty. 1155.12/300.02 Rewrite Strategy: INNERMOST 1155.35/300.10 EOF