1084.45/291.57 WORST_CASE(Omega(n^1), ?) 1084.45/291.58 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1084.45/291.58 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1084.45/291.58 1084.45/291.58 1084.45/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1084.45/291.58 1084.45/291.58 (0) CpxTRS 1084.45/291.58 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1084.45/291.58 (2) TRS for Loop Detection 1084.45/291.58 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1084.45/291.58 (4) BEST 1084.45/291.58 (5) proven lower bound 1084.45/291.58 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1084.45/291.58 (7) BOUNDS(n^1, INF) 1084.45/291.58 (8) TRS for Loop Detection 1084.45/291.58 1084.45/291.58 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (0) 1084.45/291.58 Obligation: 1084.45/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1084.45/291.58 1084.45/291.58 1084.45/291.58 The TRS R consists of the following rules: 1084.45/291.58 1084.45/291.58 a__nats -> cons(0, incr(nats)) 1084.45/291.58 a__pairs -> cons(0, incr(odds)) 1084.45/291.58 a__odds -> a__incr(a__pairs) 1084.45/291.58 a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) 1084.45/291.58 a__head(cons(X, XS)) -> mark(X) 1084.45/291.58 a__tail(cons(X, XS)) -> mark(XS) 1084.45/291.58 mark(nats) -> a__nats 1084.45/291.58 mark(pairs) -> a__pairs 1084.45/291.58 mark(odds) -> a__odds 1084.45/291.58 mark(incr(X)) -> a__incr(mark(X)) 1084.45/291.58 mark(head(X)) -> a__head(mark(X)) 1084.45/291.58 mark(tail(X)) -> a__tail(mark(X)) 1084.45/291.58 mark(0) -> 0 1084.45/291.58 mark(s(X)) -> s(mark(X)) 1084.45/291.58 mark(nil) -> nil 1084.45/291.58 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1084.45/291.58 a__nats -> nats 1084.45/291.58 a__pairs -> pairs 1084.45/291.58 a__odds -> odds 1084.45/291.58 a__incr(X) -> incr(X) 1084.45/291.58 a__head(X) -> head(X) 1084.45/291.58 a__tail(X) -> tail(X) 1084.45/291.58 1084.45/291.58 S is empty. 1084.45/291.58 Rewrite Strategy: INNERMOST 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1084.45/291.58 Transformed a relative TRS into a decreasing-loop problem. 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (2) 1084.45/291.58 Obligation: 1084.45/291.58 Analyzing the following TRS for decreasing loops: 1084.45/291.58 1084.45/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1084.45/291.58 1084.45/291.58 1084.45/291.58 The TRS R consists of the following rules: 1084.45/291.58 1084.45/291.58 a__nats -> cons(0, incr(nats)) 1084.45/291.58 a__pairs -> cons(0, incr(odds)) 1084.45/291.58 a__odds -> a__incr(a__pairs) 1084.45/291.58 a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) 1084.45/291.58 a__head(cons(X, XS)) -> mark(X) 1084.45/291.58 a__tail(cons(X, XS)) -> mark(XS) 1084.45/291.58 mark(nats) -> a__nats 1084.45/291.58 mark(pairs) -> a__pairs 1084.45/291.58 mark(odds) -> a__odds 1084.45/291.58 mark(incr(X)) -> a__incr(mark(X)) 1084.45/291.58 mark(head(X)) -> a__head(mark(X)) 1084.45/291.58 mark(tail(X)) -> a__tail(mark(X)) 1084.45/291.58 mark(0) -> 0 1084.45/291.58 mark(s(X)) -> s(mark(X)) 1084.45/291.58 mark(nil) -> nil 1084.45/291.58 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1084.45/291.58 a__nats -> nats 1084.45/291.58 a__pairs -> pairs 1084.45/291.58 a__odds -> odds 1084.45/291.58 a__incr(X) -> incr(X) 1084.45/291.58 a__head(X) -> head(X) 1084.45/291.58 a__tail(X) -> tail(X) 1084.45/291.58 1084.45/291.58 S is empty. 1084.45/291.58 Rewrite Strategy: INNERMOST 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1084.45/291.58 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1084.45/291.58 1084.45/291.58 The rewrite sequence 1084.45/291.58 1084.45/291.58 mark(incr(X)) ->^+ a__incr(mark(X)) 1084.45/291.58 1084.45/291.58 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1084.45/291.58 1084.45/291.58 The pumping substitution is [X / incr(X)]. 1084.45/291.58 1084.45/291.58 The result substitution is [ ]. 1084.45/291.58 1084.45/291.58 1084.45/291.58 1084.45/291.58 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (4) 1084.45/291.58 Complex Obligation (BEST) 1084.45/291.58 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (5) 1084.45/291.58 Obligation: 1084.45/291.58 Proved the lower bound n^1 for the following obligation: 1084.45/291.58 1084.45/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1084.45/291.58 1084.45/291.58 1084.45/291.58 The TRS R consists of the following rules: 1084.45/291.58 1084.45/291.58 a__nats -> cons(0, incr(nats)) 1084.45/291.58 a__pairs -> cons(0, incr(odds)) 1084.45/291.58 a__odds -> a__incr(a__pairs) 1084.45/291.58 a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) 1084.45/291.58 a__head(cons(X, XS)) -> mark(X) 1084.45/291.58 a__tail(cons(X, XS)) -> mark(XS) 1084.45/291.58 mark(nats) -> a__nats 1084.45/291.58 mark(pairs) -> a__pairs 1084.45/291.58 mark(odds) -> a__odds 1084.45/291.58 mark(incr(X)) -> a__incr(mark(X)) 1084.45/291.58 mark(head(X)) -> a__head(mark(X)) 1084.45/291.58 mark(tail(X)) -> a__tail(mark(X)) 1084.45/291.58 mark(0) -> 0 1084.45/291.58 mark(s(X)) -> s(mark(X)) 1084.45/291.58 mark(nil) -> nil 1084.45/291.58 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1084.45/291.58 a__nats -> nats 1084.45/291.58 a__pairs -> pairs 1084.45/291.58 a__odds -> odds 1084.45/291.58 a__incr(X) -> incr(X) 1084.45/291.58 a__head(X) -> head(X) 1084.45/291.58 a__tail(X) -> tail(X) 1084.45/291.58 1084.45/291.58 S is empty. 1084.45/291.58 Rewrite Strategy: INNERMOST 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (6) LowerBoundPropagationProof (FINISHED) 1084.45/291.58 Propagated lower bound. 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (7) 1084.45/291.58 BOUNDS(n^1, INF) 1084.45/291.58 1084.45/291.58 ---------------------------------------- 1084.45/291.58 1084.45/291.58 (8) 1084.45/291.58 Obligation: 1084.45/291.58 Analyzing the following TRS for decreasing loops: 1084.45/291.58 1084.45/291.58 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1084.45/291.58 1084.45/291.58 1084.45/291.58 The TRS R consists of the following rules: 1084.45/291.58 1084.45/291.58 a__nats -> cons(0, incr(nats)) 1084.45/291.58 a__pairs -> cons(0, incr(odds)) 1084.45/291.58 a__odds -> a__incr(a__pairs) 1084.45/291.58 a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) 1084.45/291.58 a__head(cons(X, XS)) -> mark(X) 1084.45/291.58 a__tail(cons(X, XS)) -> mark(XS) 1084.45/291.58 mark(nats) -> a__nats 1084.45/291.58 mark(pairs) -> a__pairs 1084.45/291.58 mark(odds) -> a__odds 1084.45/291.58 mark(incr(X)) -> a__incr(mark(X)) 1084.45/291.58 mark(head(X)) -> a__head(mark(X)) 1084.45/291.58 mark(tail(X)) -> a__tail(mark(X)) 1084.45/291.58 mark(0) -> 0 1084.45/291.58 mark(s(X)) -> s(mark(X)) 1084.45/291.58 mark(nil) -> nil 1084.45/291.58 mark(cons(X1, X2)) -> cons(mark(X1), X2) 1084.45/291.58 a__nats -> nats 1084.45/291.58 a__pairs -> pairs 1084.45/291.58 a__odds -> odds 1084.45/291.58 a__incr(X) -> incr(X) 1084.45/291.58 a__head(X) -> head(X) 1084.45/291.58 a__tail(X) -> tail(X) 1084.45/291.58 1084.45/291.58 S is empty. 1084.45/291.58 Rewrite Strategy: INNERMOST 1084.66/291.64 EOF