1124.01/291.53 WORST_CASE(Omega(n^1), ?) 1124.01/291.55 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1124.01/291.55 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1124.01/291.55 1124.01/291.55 1124.01/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1124.01/291.55 1124.01/291.55 (0) CpxTRS 1124.01/291.55 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1124.01/291.55 (2) TRS for Loop Detection 1124.01/291.55 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1124.01/291.55 (4) BEST 1124.01/291.55 (5) proven lower bound 1124.01/291.55 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1124.01/291.55 (7) BOUNDS(n^1, INF) 1124.01/291.55 (8) TRS for Loop Detection 1124.01/291.55 1124.01/291.55 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (0) 1124.01/291.55 Obligation: 1124.01/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1124.01/291.55 1124.01/291.55 1124.01/291.55 The TRS R consists of the following rules: 1124.01/291.55 1124.01/291.55 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 1124.01/291.55 a__sqr(0) -> 0 1124.01/291.55 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 1124.01/291.55 a__dbl(0) -> 0 1124.01/291.55 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 1124.01/291.55 a__add(0, X) -> mark(X) 1124.01/291.55 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 1124.01/291.55 a__first(0, X) -> nil 1124.01/291.55 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1124.01/291.55 a__half(0) -> 0 1124.01/291.55 a__half(s(0)) -> 0 1124.01/291.55 a__half(s(s(X))) -> s(a__half(mark(X))) 1124.01/291.55 a__half(dbl(X)) -> mark(X) 1124.01/291.55 mark(terms(X)) -> a__terms(mark(X)) 1124.01/291.55 mark(sqr(X)) -> a__sqr(mark(X)) 1124.01/291.55 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 1124.01/291.55 mark(dbl(X)) -> a__dbl(mark(X)) 1124.01/291.55 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1124.01/291.55 mark(half(X)) -> a__half(mark(X)) 1124.01/291.55 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1124.01/291.55 mark(recip(X)) -> recip(mark(X)) 1124.01/291.55 mark(s(X)) -> s(mark(X)) 1124.01/291.55 mark(0) -> 0 1124.01/291.55 mark(nil) -> nil 1124.01/291.55 a__terms(X) -> terms(X) 1124.01/291.55 a__sqr(X) -> sqr(X) 1124.01/291.55 a__add(X1, X2) -> add(X1, X2) 1124.01/291.55 a__dbl(X) -> dbl(X) 1124.01/291.55 a__first(X1, X2) -> first(X1, X2) 1124.01/291.55 a__half(X) -> half(X) 1124.01/291.55 1124.01/291.55 S is empty. 1124.01/291.55 Rewrite Strategy: INNERMOST 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1124.01/291.55 Transformed a relative TRS into a decreasing-loop problem. 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (2) 1124.01/291.55 Obligation: 1124.01/291.55 Analyzing the following TRS for decreasing loops: 1124.01/291.55 1124.01/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1124.01/291.55 1124.01/291.55 1124.01/291.55 The TRS R consists of the following rules: 1124.01/291.55 1124.01/291.55 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 1124.01/291.55 a__sqr(0) -> 0 1124.01/291.55 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 1124.01/291.55 a__dbl(0) -> 0 1124.01/291.55 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 1124.01/291.55 a__add(0, X) -> mark(X) 1124.01/291.55 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 1124.01/291.55 a__first(0, X) -> nil 1124.01/291.55 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1124.01/291.55 a__half(0) -> 0 1124.01/291.55 a__half(s(0)) -> 0 1124.01/291.55 a__half(s(s(X))) -> s(a__half(mark(X))) 1124.01/291.55 a__half(dbl(X)) -> mark(X) 1124.01/291.55 mark(terms(X)) -> a__terms(mark(X)) 1124.01/291.55 mark(sqr(X)) -> a__sqr(mark(X)) 1124.01/291.55 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 1124.01/291.55 mark(dbl(X)) -> a__dbl(mark(X)) 1124.01/291.55 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1124.01/291.55 mark(half(X)) -> a__half(mark(X)) 1124.01/291.55 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1124.01/291.55 mark(recip(X)) -> recip(mark(X)) 1124.01/291.55 mark(s(X)) -> s(mark(X)) 1124.01/291.55 mark(0) -> 0 1124.01/291.55 mark(nil) -> nil 1124.01/291.55 a__terms(X) -> terms(X) 1124.01/291.55 a__sqr(X) -> sqr(X) 1124.01/291.55 a__add(X1, X2) -> add(X1, X2) 1124.01/291.55 a__dbl(X) -> dbl(X) 1124.01/291.55 a__first(X1, X2) -> first(X1, X2) 1124.01/291.55 a__half(X) -> half(X) 1124.01/291.55 1124.01/291.55 S is empty. 1124.01/291.55 Rewrite Strategy: INNERMOST 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1124.01/291.55 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1124.01/291.55 1124.01/291.55 The rewrite sequence 1124.01/291.55 1124.01/291.55 mark(terms(X)) ->^+ a__terms(mark(X)) 1124.01/291.55 1124.01/291.55 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1124.01/291.55 1124.01/291.55 The pumping substitution is [X / terms(X)]. 1124.01/291.55 1124.01/291.55 The result substitution is [ ]. 1124.01/291.55 1124.01/291.55 1124.01/291.55 1124.01/291.55 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (4) 1124.01/291.55 Complex Obligation (BEST) 1124.01/291.55 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (5) 1124.01/291.55 Obligation: 1124.01/291.55 Proved the lower bound n^1 for the following obligation: 1124.01/291.55 1124.01/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1124.01/291.55 1124.01/291.55 1124.01/291.55 The TRS R consists of the following rules: 1124.01/291.55 1124.01/291.55 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 1124.01/291.55 a__sqr(0) -> 0 1124.01/291.55 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 1124.01/291.55 a__dbl(0) -> 0 1124.01/291.55 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 1124.01/291.55 a__add(0, X) -> mark(X) 1124.01/291.55 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 1124.01/291.55 a__first(0, X) -> nil 1124.01/291.55 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1124.01/291.55 a__half(0) -> 0 1124.01/291.55 a__half(s(0)) -> 0 1124.01/291.55 a__half(s(s(X))) -> s(a__half(mark(X))) 1124.01/291.55 a__half(dbl(X)) -> mark(X) 1124.01/291.55 mark(terms(X)) -> a__terms(mark(X)) 1124.01/291.55 mark(sqr(X)) -> a__sqr(mark(X)) 1124.01/291.55 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 1124.01/291.55 mark(dbl(X)) -> a__dbl(mark(X)) 1124.01/291.55 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1124.01/291.55 mark(half(X)) -> a__half(mark(X)) 1124.01/291.55 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1124.01/291.55 mark(recip(X)) -> recip(mark(X)) 1124.01/291.55 mark(s(X)) -> s(mark(X)) 1124.01/291.55 mark(0) -> 0 1124.01/291.55 mark(nil) -> nil 1124.01/291.55 a__terms(X) -> terms(X) 1124.01/291.55 a__sqr(X) -> sqr(X) 1124.01/291.55 a__add(X1, X2) -> add(X1, X2) 1124.01/291.55 a__dbl(X) -> dbl(X) 1124.01/291.55 a__first(X1, X2) -> first(X1, X2) 1124.01/291.55 a__half(X) -> half(X) 1124.01/291.55 1124.01/291.55 S is empty. 1124.01/291.55 Rewrite Strategy: INNERMOST 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (6) LowerBoundPropagationProof (FINISHED) 1124.01/291.55 Propagated lower bound. 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (7) 1124.01/291.55 BOUNDS(n^1, INF) 1124.01/291.55 1124.01/291.55 ---------------------------------------- 1124.01/291.55 1124.01/291.55 (8) 1124.01/291.55 Obligation: 1124.01/291.55 Analyzing the following TRS for decreasing loops: 1124.01/291.55 1124.01/291.55 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1124.01/291.55 1124.01/291.55 1124.01/291.55 The TRS R consists of the following rules: 1124.01/291.55 1124.01/291.55 a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 1124.01/291.55 a__sqr(0) -> 0 1124.01/291.55 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) 1124.01/291.55 a__dbl(0) -> 0 1124.01/291.55 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) 1124.01/291.55 a__add(0, X) -> mark(X) 1124.01/291.55 a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) 1124.01/291.55 a__first(0, X) -> nil 1124.01/291.55 a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 1124.01/291.55 a__half(0) -> 0 1124.01/291.55 a__half(s(0)) -> 0 1124.01/291.55 a__half(s(s(X))) -> s(a__half(mark(X))) 1124.01/291.55 a__half(dbl(X)) -> mark(X) 1124.01/291.55 mark(terms(X)) -> a__terms(mark(X)) 1124.01/291.55 mark(sqr(X)) -> a__sqr(mark(X)) 1124.01/291.55 mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 1124.01/291.55 mark(dbl(X)) -> a__dbl(mark(X)) 1124.01/291.55 mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 1124.01/291.55 mark(half(X)) -> a__half(mark(X)) 1124.01/291.55 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1124.01/291.55 mark(recip(X)) -> recip(mark(X)) 1124.01/291.55 mark(s(X)) -> s(mark(X)) 1124.01/291.55 mark(0) -> 0 1124.01/291.55 mark(nil) -> nil 1124.01/291.55 a__terms(X) -> terms(X) 1124.01/291.55 a__sqr(X) -> sqr(X) 1124.01/291.55 a__add(X1, X2) -> add(X1, X2) 1124.01/291.55 a__dbl(X) -> dbl(X) 1124.01/291.55 a__first(X1, X2) -> first(X1, X2) 1124.01/291.55 a__half(X) -> half(X) 1124.01/291.55 1124.01/291.55 S is empty. 1124.01/291.55 Rewrite Strategy: INNERMOST 1124.26/291.62 EOF