6.91/2.56 WORST_CASE(NON_POLY, ?) 7.27/2.57 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 7.27/2.57 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 7.27/2.57 7.27/2.57 7.27/2.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 7.27/2.57 7.27/2.57 (0) CpxTRS 7.27/2.57 (1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 7.27/2.57 (2) TRS for Loop Detection 7.27/2.57 (3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 7.27/2.57 (4) BEST 7.27/2.57 (5) proven lower bound 7.27/2.57 (6) LowerBoundPropagationProof [FINISHED, 0 ms] 7.27/2.57 (7) BOUNDS(n^1, INF) 7.27/2.57 (8) TRS for Loop Detection 7.27/2.57 (9) InfiniteLowerBoundProof [FINISHED, 652 ms] 7.27/2.57 (10) BOUNDS(INF, INF) 7.27/2.57 7.27/2.57 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (0) 7.27/2.57 Obligation: 7.27/2.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 7.27/2.57 7.27/2.57 7.27/2.57 The TRS R consists of the following rules: 7.27/2.57 7.27/2.57 nats -> adx(zeros) 7.27/2.57 zeros -> cons(n__0, n__zeros) 7.27/2.57 incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) 7.27/2.57 adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) 7.27/2.57 hd(cons(X, Y)) -> activate(X) 7.27/2.57 tl(cons(X, Y)) -> activate(Y) 7.27/2.57 0 -> n__0 7.27/2.57 zeros -> n__zeros 7.27/2.57 s(X) -> n__s(X) 7.27/2.57 incr(X) -> n__incr(X) 7.27/2.57 adx(X) -> n__adx(X) 7.27/2.57 activate(n__0) -> 0 7.27/2.57 activate(n__zeros) -> zeros 7.27/2.57 activate(n__s(X)) -> s(X) 7.27/2.57 activate(n__incr(X)) -> incr(X) 7.27/2.57 activate(n__adx(X)) -> adx(X) 7.27/2.57 activate(X) -> X 7.27/2.57 7.27/2.57 S is empty. 7.27/2.57 Rewrite Strategy: INNERMOST 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 7.27/2.57 Transformed a relative TRS into a decreasing-loop problem. 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (2) 7.27/2.57 Obligation: 7.27/2.57 Analyzing the following TRS for decreasing loops: 7.27/2.57 7.27/2.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 7.27/2.57 7.27/2.57 7.27/2.57 The TRS R consists of the following rules: 7.27/2.57 7.27/2.57 nats -> adx(zeros) 7.27/2.57 zeros -> cons(n__0, n__zeros) 7.27/2.57 incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) 7.27/2.57 adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) 7.27/2.57 hd(cons(X, Y)) -> activate(X) 7.27/2.57 tl(cons(X, Y)) -> activate(Y) 7.27/2.57 0 -> n__0 7.27/2.57 zeros -> n__zeros 7.27/2.57 s(X) -> n__s(X) 7.27/2.57 incr(X) -> n__incr(X) 7.27/2.57 adx(X) -> n__adx(X) 7.27/2.57 activate(n__0) -> 0 7.27/2.57 activate(n__zeros) -> zeros 7.27/2.57 activate(n__s(X)) -> s(X) 7.27/2.57 activate(n__incr(X)) -> incr(X) 7.27/2.57 activate(n__adx(X)) -> adx(X) 7.27/2.57 activate(X) -> X 7.27/2.57 7.27/2.57 S is empty. 7.27/2.57 Rewrite Strategy: INNERMOST 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (3) DecreasingLoopProof (LOWER BOUND(ID)) 7.27/2.57 The following loop(s) give(s) rise to the lower bound Omega(n^1): 7.27/2.57 7.27/2.57 The rewrite sequence 7.27/2.57 7.27/2.57 activate(n__incr(cons(X1_0, Y2_0))) ->^+ cons(n__s(activate(X1_0)), n__incr(activate(Y2_0))) 7.27/2.57 7.27/2.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. 7.27/2.57 7.27/2.57 The pumping substitution is [X1_0 / n__incr(cons(X1_0, Y2_0))]. 7.27/2.57 7.27/2.57 The result substitution is [ ]. 7.27/2.57 7.27/2.57 7.27/2.57 7.27/2.57 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (4) 7.27/2.57 Complex Obligation (BEST) 7.27/2.57 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (5) 7.27/2.57 Obligation: 7.27/2.57 Proved the lower bound n^1 for the following obligation: 7.27/2.57 7.27/2.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 7.27/2.57 7.27/2.57 7.27/2.57 The TRS R consists of the following rules: 7.27/2.57 7.27/2.57 nats -> adx(zeros) 7.27/2.57 zeros -> cons(n__0, n__zeros) 7.27/2.57 incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) 7.27/2.57 adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) 7.27/2.57 hd(cons(X, Y)) -> activate(X) 7.27/2.57 tl(cons(X, Y)) -> activate(Y) 7.27/2.57 0 -> n__0 7.27/2.57 zeros -> n__zeros 7.27/2.57 s(X) -> n__s(X) 7.27/2.57 incr(X) -> n__incr(X) 7.27/2.57 adx(X) -> n__adx(X) 7.27/2.57 activate(n__0) -> 0 7.27/2.57 activate(n__zeros) -> zeros 7.27/2.57 activate(n__s(X)) -> s(X) 7.27/2.57 activate(n__incr(X)) -> incr(X) 7.27/2.57 activate(n__adx(X)) -> adx(X) 7.27/2.57 activate(X) -> X 7.27/2.57 7.27/2.57 S is empty. 7.27/2.57 Rewrite Strategy: INNERMOST 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (6) LowerBoundPropagationProof (FINISHED) 7.27/2.57 Propagated lower bound. 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (7) 7.27/2.57 BOUNDS(n^1, INF) 7.27/2.57 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (8) 7.27/2.57 Obligation: 7.27/2.57 Analyzing the following TRS for decreasing loops: 7.27/2.57 7.27/2.57 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(INF, INF). 7.27/2.57 7.27/2.57 7.27/2.57 The TRS R consists of the following rules: 7.27/2.57 7.27/2.57 nats -> adx(zeros) 7.27/2.57 zeros -> cons(n__0, n__zeros) 7.27/2.57 incr(cons(X, Y)) -> cons(n__s(activate(X)), n__incr(activate(Y))) 7.27/2.57 adx(cons(X, Y)) -> incr(cons(activate(X), n__adx(activate(Y)))) 7.27/2.57 hd(cons(X, Y)) -> activate(X) 7.27/2.57 tl(cons(X, Y)) -> activate(Y) 7.27/2.57 0 -> n__0 7.27/2.57 zeros -> n__zeros 7.27/2.57 s(X) -> n__s(X) 7.27/2.57 incr(X) -> n__incr(X) 7.27/2.57 adx(X) -> n__adx(X) 7.27/2.57 activate(n__0) -> 0 7.27/2.57 activate(n__zeros) -> zeros 7.27/2.57 activate(n__s(X)) -> s(X) 7.27/2.57 activate(n__incr(X)) -> incr(X) 7.27/2.57 activate(n__adx(X)) -> adx(X) 7.27/2.57 activate(X) -> X 7.27/2.57 7.27/2.57 S is empty. 7.27/2.57 Rewrite Strategy: INNERMOST 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (9) InfiniteLowerBoundProof (FINISHED) 7.27/2.57 The following loop proves infinite runtime complexity: 7.27/2.57 7.27/2.57 The rewrite sequence 7.27/2.57 7.27/2.57 incr(cons(X, n__adx(cons(X1_1, n__zeros)))) ->^+ cons(n__s(activate(X)), n__incr(incr(cons(activate(X1_1), n__adx(cons(n__0, n__zeros)))))) 7.27/2.57 7.27/2.57 gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0]. 7.27/2.57 7.27/2.57 The pumping substitution is [ ]. 7.27/2.57 7.27/2.57 The result substitution is [X / activate(X1_1), X1_1 / n__0]. 7.27/2.57 7.27/2.57 7.27/2.57 7.27/2.57 7.27/2.57 ---------------------------------------- 7.27/2.57 7.27/2.57 (10) 7.27/2.57 BOUNDS(INF, INF) 7.29/2.59 EOF