1065.68/291.56 WORST_CASE(Omega(n^1), ?) 1065.68/291.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1065.68/291.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1065.68/291.57 1065.68/291.57 1065.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1065.68/291.57 1065.68/291.57 (0) CpxTRS 1065.68/291.57 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1065.68/291.57 (2) TRS for Loop Detection 1065.68/291.57 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1065.68/291.57 (4) BEST 1065.68/291.57 (5) proven lower bound 1065.68/291.57 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 1065.68/291.57 (7) BOUNDS(n^1, INF) 1065.68/291.57 (8) TRS for Loop Detection 1065.68/291.57 1065.68/291.57 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (0) 1065.68/291.57 Obligation: 1065.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1065.68/291.57 1065.68/291.57 1065.68/291.57 The TRS R consists of the following rules: 1065.68/291.57 1065.68/291.57 a__nats -> a__adx(a__zeros) 1065.68/291.57 a__zeros -> cons(0, zeros) 1065.68/291.57 a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 1065.68/291.57 a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 1065.68/291.57 a__hd(cons(X, Y)) -> mark(X) 1065.68/291.57 a__tl(cons(X, Y)) -> mark(Y) 1065.68/291.57 mark(nats) -> a__nats 1065.68/291.57 mark(adx(X)) -> a__adx(mark(X)) 1065.68/291.57 mark(zeros) -> a__zeros 1065.68/291.57 mark(incr(X)) -> a__incr(mark(X)) 1065.68/291.57 mark(hd(X)) -> a__hd(mark(X)) 1065.68/291.57 mark(tl(X)) -> a__tl(mark(X)) 1065.68/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 1065.68/291.57 mark(0) -> 0 1065.68/291.57 mark(s(X)) -> s(X) 1065.68/291.57 a__nats -> nats 1065.68/291.57 a__adx(X) -> adx(X) 1065.68/291.57 a__zeros -> zeros 1065.68/291.57 a__incr(X) -> incr(X) 1065.68/291.57 a__hd(X) -> hd(X) 1065.68/291.57 a__tl(X) -> tl(X) 1065.68/291.57 1065.68/291.57 S is empty. 1065.68/291.57 Rewrite Strategy: INNERMOST 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1065.68/291.57 Transformed a relative TRS into a decreasing-loop problem. 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (2) 1065.68/291.57 Obligation: 1065.68/291.57 Analyzing the following TRS for decreasing loops: 1065.68/291.57 1065.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1065.68/291.57 1065.68/291.57 1065.68/291.57 The TRS R consists of the following rules: 1065.68/291.57 1065.68/291.57 a__nats -> a__adx(a__zeros) 1065.68/291.57 a__zeros -> cons(0, zeros) 1065.68/291.57 a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 1065.68/291.57 a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 1065.68/291.57 a__hd(cons(X, Y)) -> mark(X) 1065.68/291.57 a__tl(cons(X, Y)) -> mark(Y) 1065.68/291.57 mark(nats) -> a__nats 1065.68/291.57 mark(adx(X)) -> a__adx(mark(X)) 1065.68/291.57 mark(zeros) -> a__zeros 1065.68/291.57 mark(incr(X)) -> a__incr(mark(X)) 1065.68/291.57 mark(hd(X)) -> a__hd(mark(X)) 1065.68/291.57 mark(tl(X)) -> a__tl(mark(X)) 1065.68/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 1065.68/291.57 mark(0) -> 0 1065.68/291.57 mark(s(X)) -> s(X) 1065.68/291.57 a__nats -> nats 1065.68/291.57 a__adx(X) -> adx(X) 1065.68/291.57 a__zeros -> zeros 1065.68/291.57 a__incr(X) -> incr(X) 1065.68/291.57 a__hd(X) -> hd(X) 1065.68/291.57 a__tl(X) -> tl(X) 1065.68/291.57 1065.68/291.57 S is empty. 1065.68/291.57 Rewrite Strategy: INNERMOST 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (3) DecreasingLoopProof (LOWER BOUND(ID)) 1065.68/291.57 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1065.68/291.57 1065.68/291.57 The rewrite sequence 1065.68/291.57 1065.68/291.57 mark(incr(X)) ->^+ a__incr(mark(X)) 1065.68/291.57 1065.68/291.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1065.68/291.57 1065.68/291.57 The pumping substitution is [X / incr(X)]. 1065.68/291.57 1065.68/291.57 The result substitution is [ ]. 1065.68/291.57 1065.68/291.57 1065.68/291.57 1065.68/291.57 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (4) 1065.68/291.57 Complex Obligation (BEST) 1065.68/291.57 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (5) 1065.68/291.57 Obligation: 1065.68/291.57 Proved the lower bound n^1 for the following obligation: 1065.68/291.57 1065.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1065.68/291.57 1065.68/291.57 1065.68/291.57 The TRS R consists of the following rules: 1065.68/291.57 1065.68/291.57 a__nats -> a__adx(a__zeros) 1065.68/291.57 a__zeros -> cons(0, zeros) 1065.68/291.57 a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 1065.68/291.57 a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 1065.68/291.57 a__hd(cons(X, Y)) -> mark(X) 1065.68/291.57 a__tl(cons(X, Y)) -> mark(Y) 1065.68/291.57 mark(nats) -> a__nats 1065.68/291.57 mark(adx(X)) -> a__adx(mark(X)) 1065.68/291.57 mark(zeros) -> a__zeros 1065.68/291.57 mark(incr(X)) -> a__incr(mark(X)) 1065.68/291.57 mark(hd(X)) -> a__hd(mark(X)) 1065.68/291.57 mark(tl(X)) -> a__tl(mark(X)) 1065.68/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 1065.68/291.57 mark(0) -> 0 1065.68/291.57 mark(s(X)) -> s(X) 1065.68/291.57 a__nats -> nats 1065.68/291.57 a__adx(X) -> adx(X) 1065.68/291.57 a__zeros -> zeros 1065.68/291.57 a__incr(X) -> incr(X) 1065.68/291.57 a__hd(X) -> hd(X) 1065.68/291.57 a__tl(X) -> tl(X) 1065.68/291.57 1065.68/291.57 S is empty. 1065.68/291.57 Rewrite Strategy: INNERMOST 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (6) LowerBoundPropagationProof (FINISHED) 1065.68/291.57 Propagated lower bound. 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (7) 1065.68/291.57 BOUNDS(n^1, INF) 1065.68/291.57 1065.68/291.57 ---------------------------------------- 1065.68/291.57 1065.68/291.57 (8) 1065.68/291.57 Obligation: 1065.68/291.57 Analyzing the following TRS for decreasing loops: 1065.68/291.57 1065.68/291.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 1065.68/291.57 1065.68/291.57 1065.68/291.57 The TRS R consists of the following rules: 1065.68/291.57 1065.68/291.57 a__nats -> a__adx(a__zeros) 1065.68/291.57 a__zeros -> cons(0, zeros) 1065.68/291.57 a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 1065.68/291.57 a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 1065.68/291.57 a__hd(cons(X, Y)) -> mark(X) 1065.68/291.57 a__tl(cons(X, Y)) -> mark(Y) 1065.68/291.57 mark(nats) -> a__nats 1065.68/291.57 mark(adx(X)) -> a__adx(mark(X)) 1065.68/291.57 mark(zeros) -> a__zeros 1065.68/291.57 mark(incr(X)) -> a__incr(mark(X)) 1065.68/291.57 mark(hd(X)) -> a__hd(mark(X)) 1065.68/291.57 mark(tl(X)) -> a__tl(mark(X)) 1065.68/291.57 mark(cons(X1, X2)) -> cons(X1, X2) 1065.68/291.57 mark(0) -> 0 1065.68/291.57 mark(s(X)) -> s(X) 1065.68/291.57 a__nats -> nats 1065.68/291.57 a__adx(X) -> adx(X) 1065.68/291.57 a__zeros -> zeros 1065.68/291.57 a__incr(X) -> incr(X) 1065.68/291.57 a__hd(X) -> hd(X) 1065.68/291.57 a__tl(X) -> tl(X) 1065.68/291.57 1065.68/291.57 S is empty. 1065.68/291.57 Rewrite Strategy: INNERMOST 1066.04/291.64 EOF