1123.80/291.49 WORST_CASE(Omega(n^1), ?) 1123.80/291.49 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1123.80/291.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1123.80/291.49 1123.80/291.49 1123.80/291.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.80/291.49 1123.80/291.49 (0) CpxTRS 1123.80/291.49 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1123.80/291.49 (2) TRS for Loop Detection 1123.80/291.49 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1123.80/291.49 (4) BEST 1123.80/291.49 (5) proven lower bound 1123.80/291.49 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1123.80/291.49 (7) BOUNDS(n^1, INF) 1123.80/291.49 (8) TRS for Loop Detection 1123.80/291.49 1123.80/291.49 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (0) 1123.80/291.49 Obligation: 1123.80/291.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.80/291.49 1123.80/291.49 1123.80/291.49 The TRS R consists of the following rules: 1123.80/291.49 1123.80/291.49 a__from(X) -> cons(mark(X), from(s(X))) 1123.80/291.49 a__first(0, Z) -> nil 1123.80/291.49 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1123.80/291.49 a__sel(0, cons(X, Z)) -> mark(X) 1123.80/291.49 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1123.80/291.49 mark(from(X)) -> a__from(mark(X)) 1123.80/291.49 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1123.80/291.49 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1123.80/291.49 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.80/291.49 mark(s(X)) -> s(mark(X)) 1123.80/291.49 mark(0) -> 0 1123.80/291.49 mark(nil) -> nil 1123.80/291.49 a__from(X) -> from(X) 1123.80/291.49 a__first(X1, X2) -> first(X1, X2) 1123.80/291.49 a__sel(X1, X2) -> sel(X1, X2) 1123.80/291.49 1123.80/291.49 S is empty. 1123.80/291.49 Rewrite Strategy: INNERMOST 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1123.80/291.49 Transformed a relative TRS into a decreasing-loop problem. 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (2) 1123.80/291.49 Obligation: 1123.80/291.49 Analyzing the following TRS for decreasing loops: 1123.80/291.49 1123.80/291.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.80/291.49 1123.80/291.49 1123.80/291.49 The TRS R consists of the following rules: 1123.80/291.49 1123.80/291.49 a__from(X) -> cons(mark(X), from(s(X))) 1123.80/291.49 a__first(0, Z) -> nil 1123.80/291.49 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1123.80/291.49 a__sel(0, cons(X, Z)) -> mark(X) 1123.80/291.49 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1123.80/291.49 mark(from(X)) -> a__from(mark(X)) 1123.80/291.49 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1123.80/291.49 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1123.80/291.49 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.80/291.49 mark(s(X)) -> s(mark(X)) 1123.80/291.49 mark(0) -> 0 1123.80/291.49 mark(nil) -> nil 1123.80/291.49 a__from(X) -> from(X) 1123.80/291.49 a__first(X1, X2) -> first(X1, X2) 1123.80/291.49 a__sel(X1, X2) -> sel(X1, X2) 1123.80/291.49 1123.80/291.49 S is empty. 1123.80/291.49 Rewrite Strategy: INNERMOST 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1123.80/291.49 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1123.80/291.49 1123.80/291.49 The rewrite sequence 1123.80/291.49 1123.80/291.49 mark(from(X)) ->^+ a__from(mark(X)) 1123.80/291.49 1123.80/291.49 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1123.80/291.49 1123.80/291.49 The pumping substitution is [X / from(X)]. 1123.80/291.49 1123.80/291.49 The result substitution is [ ]. 1123.80/291.49 1123.80/291.49 1123.80/291.49 1123.80/291.49 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (4) 1123.80/291.49 Complex Obligation (BEST) 1123.80/291.49 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (5) 1123.80/291.49 Obligation: 1123.80/291.49 Proved the lower bound n^1 for the following obligation: 1123.80/291.49 1123.80/291.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.80/291.49 1123.80/291.49 1123.80/291.49 The TRS R consists of the following rules: 1123.80/291.49 1123.80/291.49 a__from(X) -> cons(mark(X), from(s(X))) 1123.80/291.49 a__first(0, Z) -> nil 1123.80/291.49 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1123.80/291.49 a__sel(0, cons(X, Z)) -> mark(X) 1123.80/291.49 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1123.80/291.49 mark(from(X)) -> a__from(mark(X)) 1123.80/291.49 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1123.80/291.49 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1123.80/291.49 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.80/291.49 mark(s(X)) -> s(mark(X)) 1123.80/291.49 mark(0) -> 0 1123.80/291.49 mark(nil) -> nil 1123.80/291.49 a__from(X) -> from(X) 1123.80/291.49 a__first(X1, X2) -> first(X1, X2) 1123.80/291.49 a__sel(X1, X2) -> sel(X1, X2) 1123.80/291.49 1123.80/291.49 S is empty. 1123.80/291.49 Rewrite Strategy: INNERMOST 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (6) LowerBoundPropagationProof (FINISHED) 1123.80/291.49 Propagated lower bound. 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (7) 1123.80/291.49 BOUNDS(n^1, INF) 1123.80/291.49 1123.80/291.49 ---------------------------------------- 1123.80/291.49 1123.80/291.49 (8) 1123.80/291.49 Obligation: 1123.80/291.49 Analyzing the following TRS for decreasing loops: 1123.80/291.49 1123.80/291.49 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1123.80/291.49 1123.80/291.49 1123.80/291.49 The TRS R consists of the following rules: 1123.80/291.49 1123.80/291.49 a__from(X) -> cons(mark(X), from(s(X))) 1123.80/291.49 a__first(0, Z) -> nil 1123.80/291.49 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1123.80/291.49 a__sel(0, cons(X, Z)) -> mark(X) 1123.80/291.49 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1123.80/291.49 mark(from(X)) -> a__from(mark(X)) 1123.80/291.49 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1123.80/291.49 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1123.80/291.49 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1123.80/291.49 mark(s(X)) -> s(mark(X)) 1123.80/291.49 mark(0) -> 0 1123.80/291.49 mark(nil) -> nil 1123.80/291.49 a__from(X) -> from(X) 1123.80/291.49 a__first(X1, X2) -> first(X1, X2) 1123.80/291.49 a__sel(X1, X2) -> sel(X1, X2) 1123.80/291.49 1123.80/291.49 S is empty. 1123.80/291.49 Rewrite Strategy: INNERMOST 1123.95/291.56 EOF