1139.03/291.54 WORST_CASE(Omega(n^1), ?) 1143.06/292.57 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1143.06/292.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1143.06/292.57 1143.06/292.57 1143.06/292.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1143.06/292.57 1143.06/292.57 (0) CpxTRS 1143.06/292.57 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1143.06/292.57 (2) TRS for Loop Detection 1143.06/292.57 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1143.06/292.57 (4) BEST 1143.06/292.57 (5) proven lower bound 1143.06/292.57 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1143.06/292.57 (7) BOUNDS(n^1, INF) 1143.06/292.57 (8) TRS for Loop Detection 1143.06/292.57 1143.06/292.57 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (0) 1143.06/292.57 Obligation: 1143.06/292.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1143.06/292.57 1143.06/292.57 1143.06/292.57 The TRS R consists of the following rules: 1143.06/292.57 1143.06/292.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1143.06/292.57 a__sel(0, cons(X, Z)) -> mark(X) 1143.06/292.57 a__first(0, Z) -> nil 1143.06/292.57 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1143.06/292.57 a__from(X) -> cons(mark(X), from(s(X))) 1143.06/292.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__sel1(0, cons(X, Z)) -> a__quote(X) 1143.06/292.57 a__first1(0, Z) -> nil1 1143.06/292.57 a__first1(s(X), cons(Y, Z)) -> cons1(a__quote(Y), a__first1(mark(X), mark(Z))) 1143.06/292.57 a__quote(0) -> 01 1143.06/292.57 a__quote1(cons(X, Z)) -> cons1(a__quote(X), a__quote1(Z)) 1143.06/292.57 a__quote1(nil) -> nil1 1143.06/292.57 a__quote(s(X)) -> s1(a__quote(X)) 1143.06/292.57 a__quote(sel(X, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__quote1(first(X, Z)) -> a__first1(mark(X), mark(Z)) 1143.06/292.57 a__unquote(01) -> 0 1143.06/292.57 a__unquote(s1(X)) -> s(a__unquote(mark(X))) 1143.06/292.57 a__unquote1(nil1) -> nil 1143.06/292.57 a__unquote1(cons1(X, Z)) -> a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z))) 1143.06/292.57 a__fcons(X, Z) -> cons(mark(X), Z) 1143.06/292.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1143.06/292.57 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1143.06/292.57 mark(from(X)) -> a__from(mark(X)) 1143.06/292.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote(X)) -> a__quote(X) 1143.06/292.57 mark(first1(X1, X2)) -> a__first1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote1(X)) -> a__quote1(X) 1143.06/292.57 mark(unquote(X)) -> a__unquote(mark(X)) 1143.06/292.57 mark(unquote1(X)) -> a__unquote1(mark(X)) 1143.06/292.57 mark(fcons(X1, X2)) -> a__fcons(mark(X1), mark(X2)) 1143.06/292.57 mark(s(X)) -> s(mark(X)) 1143.06/292.57 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1143.06/292.57 mark(0) -> 0 1143.06/292.57 mark(nil) -> nil 1143.06/292.57 mark(nil1) -> nil1 1143.06/292.57 mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) 1143.06/292.57 mark(01) -> 01 1143.06/292.57 mark(s1(X)) -> s1(mark(X)) 1143.06/292.57 a__sel(X1, X2) -> sel(X1, X2) 1143.06/292.57 a__first(X1, X2) -> first(X1, X2) 1143.06/292.57 a__from(X) -> from(X) 1143.06/292.57 a__sel1(X1, X2) -> sel1(X1, X2) 1143.06/292.57 a__quote(X) -> quote(X) 1143.06/292.57 a__first1(X1, X2) -> first1(X1, X2) 1143.06/292.57 a__quote1(X) -> quote1(X) 1143.06/292.57 a__unquote(X) -> unquote(X) 1143.06/292.57 a__unquote1(X) -> unquote1(X) 1143.06/292.57 a__fcons(X1, X2) -> fcons(X1, X2) 1143.06/292.57 1143.06/292.57 S is empty. 1143.06/292.57 Rewrite Strategy: INNERMOST 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1143.06/292.57 Transformed a relative TRS into a decreasing-loop problem. 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (2) 1143.06/292.57 Obligation: 1143.06/292.57 Analyzing the following TRS for decreasing loops: 1143.06/292.57 1143.06/292.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1143.06/292.57 1143.06/292.57 1143.06/292.57 The TRS R consists of the following rules: 1143.06/292.57 1143.06/292.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1143.06/292.57 a__sel(0, cons(X, Z)) -> mark(X) 1143.06/292.57 a__first(0, Z) -> nil 1143.06/292.57 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1143.06/292.57 a__from(X) -> cons(mark(X), from(s(X))) 1143.06/292.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__sel1(0, cons(X, Z)) -> a__quote(X) 1143.06/292.57 a__first1(0, Z) -> nil1 1143.06/292.57 a__first1(s(X), cons(Y, Z)) -> cons1(a__quote(Y), a__first1(mark(X), mark(Z))) 1143.06/292.57 a__quote(0) -> 01 1143.06/292.57 a__quote1(cons(X, Z)) -> cons1(a__quote(X), a__quote1(Z)) 1143.06/292.57 a__quote1(nil) -> nil1 1143.06/292.57 a__quote(s(X)) -> s1(a__quote(X)) 1143.06/292.57 a__quote(sel(X, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__quote1(first(X, Z)) -> a__first1(mark(X), mark(Z)) 1143.06/292.57 a__unquote(01) -> 0 1143.06/292.57 a__unquote(s1(X)) -> s(a__unquote(mark(X))) 1143.06/292.57 a__unquote1(nil1) -> nil 1143.06/292.57 a__unquote1(cons1(X, Z)) -> a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z))) 1143.06/292.57 a__fcons(X, Z) -> cons(mark(X), Z) 1143.06/292.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1143.06/292.57 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1143.06/292.57 mark(from(X)) -> a__from(mark(X)) 1143.06/292.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote(X)) -> a__quote(X) 1143.06/292.57 mark(first1(X1, X2)) -> a__first1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote1(X)) -> a__quote1(X) 1143.06/292.57 mark(unquote(X)) -> a__unquote(mark(X)) 1143.06/292.57 mark(unquote1(X)) -> a__unquote1(mark(X)) 1143.06/292.57 mark(fcons(X1, X2)) -> a__fcons(mark(X1), mark(X2)) 1143.06/292.57 mark(s(X)) -> s(mark(X)) 1143.06/292.57 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1143.06/292.57 mark(0) -> 0 1143.06/292.57 mark(nil) -> nil 1143.06/292.57 mark(nil1) -> nil1 1143.06/292.57 mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) 1143.06/292.57 mark(01) -> 01 1143.06/292.57 mark(s1(X)) -> s1(mark(X)) 1143.06/292.57 a__sel(X1, X2) -> sel(X1, X2) 1143.06/292.57 a__first(X1, X2) -> first(X1, X2) 1143.06/292.57 a__from(X) -> from(X) 1143.06/292.57 a__sel1(X1, X2) -> sel1(X1, X2) 1143.06/292.57 a__quote(X) -> quote(X) 1143.06/292.57 a__first1(X1, X2) -> first1(X1, X2) 1143.06/292.57 a__quote1(X) -> quote1(X) 1143.06/292.57 a__unquote(X) -> unquote(X) 1143.06/292.57 a__unquote1(X) -> unquote1(X) 1143.06/292.57 a__fcons(X1, X2) -> fcons(X1, X2) 1143.06/292.57 1143.06/292.57 S is empty. 1143.06/292.57 Rewrite Strategy: INNERMOST 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1143.06/292.57 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1143.06/292.57 1143.06/292.57 The rewrite sequence 1143.06/292.57 1143.06/292.57 mark(sel1(X1, X2)) ->^+ a__sel1(mark(X1), mark(X2)) 1143.06/292.57 1143.06/292.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1143.06/292.57 1143.06/292.57 The pumping substitution is [X1 / sel1(X1, X2)]. 1143.06/292.57 1143.06/292.57 The result substitution is [ ]. 1143.06/292.57 1143.06/292.57 1143.06/292.57 1143.06/292.57 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (4) 1143.06/292.57 Complex Obligation (BEST) 1143.06/292.57 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (5) 1143.06/292.57 Obligation: 1143.06/292.57 Proved the lower bound n^1 for the following obligation: 1143.06/292.57 1143.06/292.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1143.06/292.57 1143.06/292.57 1143.06/292.57 The TRS R consists of the following rules: 1143.06/292.57 1143.06/292.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1143.06/292.57 a__sel(0, cons(X, Z)) -> mark(X) 1143.06/292.57 a__first(0, Z) -> nil 1143.06/292.57 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1143.06/292.57 a__from(X) -> cons(mark(X), from(s(X))) 1143.06/292.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__sel1(0, cons(X, Z)) -> a__quote(X) 1143.06/292.57 a__first1(0, Z) -> nil1 1143.06/292.57 a__first1(s(X), cons(Y, Z)) -> cons1(a__quote(Y), a__first1(mark(X), mark(Z))) 1143.06/292.57 a__quote(0) -> 01 1143.06/292.57 a__quote1(cons(X, Z)) -> cons1(a__quote(X), a__quote1(Z)) 1143.06/292.57 a__quote1(nil) -> nil1 1143.06/292.57 a__quote(s(X)) -> s1(a__quote(X)) 1143.06/292.57 a__quote(sel(X, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__quote1(first(X, Z)) -> a__first1(mark(X), mark(Z)) 1143.06/292.57 a__unquote(01) -> 0 1143.06/292.57 a__unquote(s1(X)) -> s(a__unquote(mark(X))) 1143.06/292.57 a__unquote1(nil1) -> nil 1143.06/292.57 a__unquote1(cons1(X, Z)) -> a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z))) 1143.06/292.57 a__fcons(X, Z) -> cons(mark(X), Z) 1143.06/292.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1143.06/292.57 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1143.06/292.57 mark(from(X)) -> a__from(mark(X)) 1143.06/292.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote(X)) -> a__quote(X) 1143.06/292.57 mark(first1(X1, X2)) -> a__first1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote1(X)) -> a__quote1(X) 1143.06/292.57 mark(unquote(X)) -> a__unquote(mark(X)) 1143.06/292.57 mark(unquote1(X)) -> a__unquote1(mark(X)) 1143.06/292.57 mark(fcons(X1, X2)) -> a__fcons(mark(X1), mark(X2)) 1143.06/292.57 mark(s(X)) -> s(mark(X)) 1143.06/292.57 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1143.06/292.57 mark(0) -> 0 1143.06/292.57 mark(nil) -> nil 1143.06/292.57 mark(nil1) -> nil1 1143.06/292.57 mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) 1143.06/292.57 mark(01) -> 01 1143.06/292.57 mark(s1(X)) -> s1(mark(X)) 1143.06/292.57 a__sel(X1, X2) -> sel(X1, X2) 1143.06/292.57 a__first(X1, X2) -> first(X1, X2) 1143.06/292.57 a__from(X) -> from(X) 1143.06/292.57 a__sel1(X1, X2) -> sel1(X1, X2) 1143.06/292.57 a__quote(X) -> quote(X) 1143.06/292.57 a__first1(X1, X2) -> first1(X1, X2) 1143.06/292.57 a__quote1(X) -> quote1(X) 1143.06/292.57 a__unquote(X) -> unquote(X) 1143.06/292.57 a__unquote1(X) -> unquote1(X) 1143.06/292.57 a__fcons(X1, X2) -> fcons(X1, X2) 1143.06/292.57 1143.06/292.57 S is empty. 1143.06/292.57 Rewrite Strategy: INNERMOST 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (6) LowerBoundPropagationProof (FINISHED) 1143.06/292.57 Propagated lower bound. 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (7) 1143.06/292.57 BOUNDS(n^1, INF) 1143.06/292.57 1143.06/292.57 ---------------------------------------- 1143.06/292.57 1143.06/292.57 (8) 1143.06/292.57 Obligation: 1143.06/292.57 Analyzing the following TRS for decreasing loops: 1143.06/292.57 1143.06/292.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1143.06/292.57 1143.06/292.57 1143.06/292.57 The TRS R consists of the following rules: 1143.06/292.57 1143.06/292.57 a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) 1143.06/292.57 a__sel(0, cons(X, Z)) -> mark(X) 1143.06/292.57 a__first(0, Z) -> nil 1143.06/292.57 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1143.06/292.57 a__from(X) -> cons(mark(X), from(s(X))) 1143.06/292.57 a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__sel1(0, cons(X, Z)) -> a__quote(X) 1143.06/292.57 a__first1(0, Z) -> nil1 1143.06/292.57 a__first1(s(X), cons(Y, Z)) -> cons1(a__quote(Y), a__first1(mark(X), mark(Z))) 1143.06/292.57 a__quote(0) -> 01 1143.06/292.57 a__quote1(cons(X, Z)) -> cons1(a__quote(X), a__quote1(Z)) 1143.06/292.57 a__quote1(nil) -> nil1 1143.06/292.57 a__quote(s(X)) -> s1(a__quote(X)) 1143.06/292.57 a__quote(sel(X, Z)) -> a__sel1(mark(X), mark(Z)) 1143.06/292.57 a__quote1(first(X, Z)) -> a__first1(mark(X), mark(Z)) 1143.06/292.57 a__unquote(01) -> 0 1143.06/292.57 a__unquote(s1(X)) -> s(a__unquote(mark(X))) 1143.06/292.57 a__unquote1(nil1) -> nil 1143.06/292.57 a__unquote1(cons1(X, Z)) -> a__fcons(a__unquote(mark(X)), a__unquote1(mark(Z))) 1143.06/292.57 a__fcons(X, Z) -> cons(mark(X), Z) 1143.06/292.57 mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) 1143.06/292.57 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1143.06/292.57 mark(from(X)) -> a__from(mark(X)) 1143.06/292.57 mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote(X)) -> a__quote(X) 1143.06/292.57 mark(first1(X1, X2)) -> a__first1(mark(X1), mark(X2)) 1143.06/292.57 mark(quote1(X)) -> a__quote1(X) 1143.06/292.57 mark(unquote(X)) -> a__unquote(mark(X)) 1143.06/292.57 mark(unquote1(X)) -> a__unquote1(mark(X)) 1143.06/292.57 mark(fcons(X1, X2)) -> a__fcons(mark(X1), mark(X2)) 1143.06/292.57 mark(s(X)) -> s(mark(X)) 1143.06/292.57 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1143.06/292.57 mark(0) -> 0 1143.06/292.57 mark(nil) -> nil 1143.06/292.57 mark(nil1) -> nil1 1143.06/292.57 mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) 1143.06/292.57 mark(01) -> 01 1143.06/292.57 mark(s1(X)) -> s1(mark(X)) 1143.06/292.57 a__sel(X1, X2) -> sel(X1, X2) 1143.06/292.57 a__first(X1, X2) -> first(X1, X2) 1143.06/292.57 a__from(X) -> from(X) 1143.06/292.57 a__sel1(X1, X2) -> sel1(X1, X2) 1143.06/292.57 a__quote(X) -> quote(X) 1143.06/292.57 a__first1(X1, X2) -> first1(X1, X2) 1143.06/292.57 a__quote1(X) -> quote1(X) 1143.06/292.57 a__unquote(X) -> unquote(X) 1143.06/292.57 a__unquote1(X) -> unquote1(X) 1143.06/292.57 a__fcons(X1, X2) -> fcons(X1, X2) 1143.06/292.57 1143.06/292.57 S is empty. 1143.06/292.57 Rewrite Strategy: INNERMOST 1143.28/292.64 EOF