1152.69/295.01 WORST_CASE(Omega(n^1), ?) 1152.69/295.02 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1152.69/295.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1152.69/295.02 1152.69/295.02 1152.69/295.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1152.69/295.02 1152.69/295.02 (0) CpxTRS 1152.69/295.02 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1152.69/295.02 (2) TRS for Loop Detection 1152.69/295.02 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1152.69/295.02 (4) BEST 1152.69/295.02 (5) proven lower bound 1152.69/295.02 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1152.69/295.02 (7) BOUNDS(n^1, INF) 1152.69/295.02 (8) TRS for Loop Detection 1152.69/295.02 1152.69/295.02 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (0) 1152.69/295.02 Obligation: 1152.69/295.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1152.69/295.02 1152.69/295.02 1152.69/295.02 The TRS R consists of the following rules: 1152.69/295.02 1152.69/295.02 a__from(X) -> cons(mark(X), from(s(X))) 1152.69/295.02 a__2ndspos(0, Z) -> rnil 1152.69/295.02 a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) 1152.69/295.02 a__2ndsneg(0, Z) -> rnil 1152.69/295.02 a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) 1152.69/295.02 a__pi(X) -> a__2ndspos(mark(X), a__from(0)) 1152.69/295.02 a__plus(0, Y) -> mark(Y) 1152.69/295.02 a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) 1152.69/295.02 a__times(0, Y) -> 0 1152.69/295.02 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) 1152.69/295.02 a__square(X) -> a__times(mark(X), mark(X)) 1152.69/295.02 mark(from(X)) -> a__from(mark(X)) 1152.69/295.02 mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) 1152.69/295.02 mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) 1152.69/295.02 mark(pi(X)) -> a__pi(mark(X)) 1152.69/295.02 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1152.69/295.02 mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) 1152.69/295.02 mark(square(X)) -> a__square(mark(X)) 1152.69/295.02 mark(0) -> 0 1152.69/295.02 mark(s(X)) -> s(mark(X)) 1152.69/295.02 mark(posrecip(X)) -> posrecip(mark(X)) 1152.69/295.02 mark(negrecip(X)) -> negrecip(mark(X)) 1152.69/295.02 mark(nil) -> nil 1152.69/295.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1152.69/295.02 mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) 1152.69/295.02 mark(rnil) -> rnil 1152.69/295.02 mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) 1152.69/295.02 a__from(X) -> from(X) 1152.69/295.02 a__2ndspos(X1, X2) -> 2ndspos(X1, X2) 1152.69/295.02 a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) 1152.69/295.02 a__pi(X) -> pi(X) 1152.69/295.02 a__plus(X1, X2) -> plus(X1, X2) 1152.69/295.02 a__times(X1, X2) -> times(X1, X2) 1152.69/295.02 a__square(X) -> square(X) 1152.69/295.02 1152.69/295.02 S is empty. 1152.69/295.02 Rewrite Strategy: INNERMOST 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1152.69/295.02 Transformed a relative TRS into a decreasing-loop problem. 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (2) 1152.69/295.02 Obligation: 1152.69/295.02 Analyzing the following TRS for decreasing loops: 1152.69/295.02 1152.69/295.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1152.69/295.02 1152.69/295.02 1152.69/295.02 The TRS R consists of the following rules: 1152.69/295.02 1152.69/295.02 a__from(X) -> cons(mark(X), from(s(X))) 1152.69/295.02 a__2ndspos(0, Z) -> rnil 1152.69/295.02 a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) 1152.69/295.02 a__2ndsneg(0, Z) -> rnil 1152.69/295.02 a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) 1152.69/295.02 a__pi(X) -> a__2ndspos(mark(X), a__from(0)) 1152.69/295.02 a__plus(0, Y) -> mark(Y) 1152.69/295.02 a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) 1152.69/295.02 a__times(0, Y) -> 0 1152.69/295.02 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) 1152.69/295.02 a__square(X) -> a__times(mark(X), mark(X)) 1152.69/295.02 mark(from(X)) -> a__from(mark(X)) 1152.69/295.02 mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) 1152.69/295.02 mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) 1152.69/295.02 mark(pi(X)) -> a__pi(mark(X)) 1152.69/295.02 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1152.69/295.02 mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) 1152.69/295.02 mark(square(X)) -> a__square(mark(X)) 1152.69/295.02 mark(0) -> 0 1152.69/295.02 mark(s(X)) -> s(mark(X)) 1152.69/295.02 mark(posrecip(X)) -> posrecip(mark(X)) 1152.69/295.02 mark(negrecip(X)) -> negrecip(mark(X)) 1152.69/295.02 mark(nil) -> nil 1152.69/295.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1152.69/295.02 mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) 1152.69/295.02 mark(rnil) -> rnil 1152.69/295.02 mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) 1152.69/295.02 a__from(X) -> from(X) 1152.69/295.02 a__2ndspos(X1, X2) -> 2ndspos(X1, X2) 1152.69/295.02 a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) 1152.69/295.02 a__pi(X) -> pi(X) 1152.69/295.02 a__plus(X1, X2) -> plus(X1, X2) 1152.69/295.02 a__times(X1, X2) -> times(X1, X2) 1152.69/295.02 a__square(X) -> square(X) 1152.69/295.02 1152.69/295.02 S is empty. 1152.69/295.02 Rewrite Strategy: INNERMOST 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1152.69/295.02 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1152.69/295.02 1152.69/295.02 The rewrite sequence 1152.69/295.02 1152.69/295.02 mark(2ndsneg(X1, X2)) ->^+ a__2ndsneg(mark(X1), mark(X2)) 1152.69/295.02 1152.69/295.02 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1152.69/295.02 1152.69/295.02 The pumping substitution is [X1 / 2ndsneg(X1, X2)]. 1152.69/295.02 1152.69/295.02 The result substitution is [ ]. 1152.69/295.02 1152.69/295.02 1152.69/295.02 1152.69/295.02 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (4) 1152.69/295.02 Complex Obligation (BEST) 1152.69/295.02 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (5) 1152.69/295.02 Obligation: 1152.69/295.02 Proved the lower bound n^1 for the following obligation: 1152.69/295.02 1152.69/295.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1152.69/295.02 1152.69/295.02 1152.69/295.02 The TRS R consists of the following rules: 1152.69/295.02 1152.69/295.02 a__from(X) -> cons(mark(X), from(s(X))) 1152.69/295.02 a__2ndspos(0, Z) -> rnil 1152.69/295.02 a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) 1152.69/295.02 a__2ndsneg(0, Z) -> rnil 1152.69/295.02 a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) 1152.69/295.02 a__pi(X) -> a__2ndspos(mark(X), a__from(0)) 1152.69/295.02 a__plus(0, Y) -> mark(Y) 1152.69/295.02 a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) 1152.69/295.02 a__times(0, Y) -> 0 1152.69/295.02 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) 1152.69/295.02 a__square(X) -> a__times(mark(X), mark(X)) 1152.69/295.02 mark(from(X)) -> a__from(mark(X)) 1152.69/295.02 mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) 1152.69/295.02 mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) 1152.69/295.02 mark(pi(X)) -> a__pi(mark(X)) 1152.69/295.02 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1152.69/295.02 mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) 1152.69/295.02 mark(square(X)) -> a__square(mark(X)) 1152.69/295.02 mark(0) -> 0 1152.69/295.02 mark(s(X)) -> s(mark(X)) 1152.69/295.02 mark(posrecip(X)) -> posrecip(mark(X)) 1152.69/295.02 mark(negrecip(X)) -> negrecip(mark(X)) 1152.69/295.02 mark(nil) -> nil 1152.69/295.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1152.69/295.02 mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) 1152.69/295.02 mark(rnil) -> rnil 1152.69/295.02 mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) 1152.69/295.02 a__from(X) -> from(X) 1152.69/295.02 a__2ndspos(X1, X2) -> 2ndspos(X1, X2) 1152.69/295.02 a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) 1152.69/295.02 a__pi(X) -> pi(X) 1152.69/295.02 a__plus(X1, X2) -> plus(X1, X2) 1152.69/295.02 a__times(X1, X2) -> times(X1, X2) 1152.69/295.02 a__square(X) -> square(X) 1152.69/295.02 1152.69/295.02 S is empty. 1152.69/295.02 Rewrite Strategy: INNERMOST 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (6) LowerBoundPropagationProof (FINISHED) 1152.69/295.02 Propagated lower bound. 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (7) 1152.69/295.02 BOUNDS(n^1, INF) 1152.69/295.02 1152.69/295.02 ---------------------------------------- 1152.69/295.02 1152.69/295.02 (8) 1152.69/295.02 Obligation: 1152.69/295.02 Analyzing the following TRS for decreasing loops: 1152.69/295.02 1152.69/295.02 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1152.69/295.02 1152.69/295.02 1152.69/295.02 The TRS R consists of the following rules: 1152.69/295.02 1152.69/295.02 a__from(X) -> cons(mark(X), from(s(X))) 1152.69/295.02 a__2ndspos(0, Z) -> rnil 1152.69/295.02 a__2ndspos(s(N), cons(X, Z)) -> a__2ndspos(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z))) 1152.69/295.02 a__2ndsneg(0, Z) -> rnil 1152.69/295.02 a__2ndsneg(s(N), cons(X, Z)) -> a__2ndsneg(s(mark(N)), cons2(X, mark(Z))) 1152.69/295.02 a__2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z))) 1152.69/295.02 a__pi(X) -> a__2ndspos(mark(X), a__from(0)) 1152.69/295.02 a__plus(0, Y) -> mark(Y) 1152.69/295.02 a__plus(s(X), Y) -> s(a__plus(mark(X), mark(Y))) 1152.69/295.02 a__times(0, Y) -> 0 1152.69/295.02 a__times(s(X), Y) -> a__plus(mark(Y), a__times(mark(X), mark(Y))) 1152.69/295.02 a__square(X) -> a__times(mark(X), mark(X)) 1152.69/295.02 mark(from(X)) -> a__from(mark(X)) 1152.69/295.02 mark(2ndspos(X1, X2)) -> a__2ndspos(mark(X1), mark(X2)) 1152.69/295.02 mark(2ndsneg(X1, X2)) -> a__2ndsneg(mark(X1), mark(X2)) 1152.69/295.02 mark(pi(X)) -> a__pi(mark(X)) 1152.69/295.02 mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1152.69/295.02 mark(times(X1, X2)) -> a__times(mark(X1), mark(X2)) 1152.69/295.02 mark(square(X)) -> a__square(mark(X)) 1152.69/295.02 mark(0) -> 0 1152.69/295.02 mark(s(X)) -> s(mark(X)) 1152.69/295.02 mark(posrecip(X)) -> posrecip(mark(X)) 1152.69/295.02 mark(negrecip(X)) -> negrecip(mark(X)) 1152.69/295.02 mark(nil) -> nil 1152.69/295.02 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1152.69/295.02 mark(cons2(X1, X2)) -> cons2(X1, mark(X2)) 1152.69/295.02 mark(rnil) -> rnil 1152.69/295.02 mark(rcons(X1, X2)) -> rcons(mark(X1), mark(X2)) 1152.69/295.02 a__from(X) -> from(X) 1152.69/295.02 a__2ndspos(X1, X2) -> 2ndspos(X1, X2) 1152.69/295.02 a__2ndsneg(X1, X2) -> 2ndsneg(X1, X2) 1152.69/295.02 a__pi(X) -> pi(X) 1152.69/295.02 a__plus(X1, X2) -> plus(X1, X2) 1152.69/295.02 a__times(X1, X2) -> times(X1, X2) 1152.69/295.02 a__square(X) -> square(X) 1152.69/295.02 1152.69/295.02 S is empty. 1152.69/295.02 Rewrite Strategy: INNERMOST 1152.79/295.09 EOF