3.73/1.69 WORST_CASE(Omega(n^1), O(n^1)) 4.06/1.70 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.06/1.70 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.06/1.70 4.06/1.70 4.06/1.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.70 4.06/1.70 (0) CpxTRS 4.06/1.70 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 4.06/1.70 (2) CpxTRS 4.06/1.70 (3) CpxTrsMatchBoundsTAProof [FINISHED, 106 ms] 4.06/1.70 (4) BOUNDS(1, n^1) 4.06/1.70 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 4.06/1.70 (6) TRS for Loop Detection 4.06/1.70 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 4.06/1.70 (8) BEST 4.06/1.70 (9) proven lower bound 4.06/1.70 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 4.06/1.70 (11) BOUNDS(n^1, INF) 4.06/1.70 (12) TRS for Loop Detection 4.06/1.70 4.06/1.70 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (0) 4.06/1.70 Obligation: 4.06/1.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.70 4.06/1.70 4.06/1.70 The TRS R consists of the following rules: 4.06/1.70 4.06/1.70 max(L(x)) -> x 4.06/1.70 max(N(L(0), L(y))) -> y 4.06/1.70 max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y)))) 4.06/1.70 max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z))))) 4.06/1.70 4.06/1.70 S is empty. 4.06/1.70 Rewrite Strategy: INNERMOST 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 4.06/1.70 transformed relative TRS to TRS 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (2) 4.06/1.70 Obligation: 4.06/1.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 4.06/1.70 4.06/1.70 4.06/1.70 The TRS R consists of the following rules: 4.06/1.70 4.06/1.70 max(L(x)) -> x 4.06/1.70 max(N(L(0), L(y))) -> y 4.06/1.70 max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y)))) 4.06/1.70 max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z))))) 4.06/1.70 4.06/1.70 S is empty. 4.06/1.70 Rewrite Strategy: INNERMOST 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (3) CpxTrsMatchBoundsTAProof (FINISHED) 4.06/1.70 A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 4.06/1.70 4.06/1.70 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 4.06/1.70 final states : [1] 4.06/1.70 transitions: 4.06/1.70 L0(0) -> 0 4.06/1.70 N0(0, 0) -> 0 4.06/1.70 00() -> 0 4.06/1.70 s0(0) -> 0 4.06/1.70 max0(0) -> 1 4.06/1.70 L1(0) -> 4 4.06/1.70 L1(0) -> 5 4.06/1.70 N1(4, 5) -> 3 4.06/1.70 max1(3) -> 2 4.06/1.70 s1(2) -> 1 4.06/1.70 N1(0, 0) -> 8 4.06/1.70 max1(8) -> 7 4.06/1.70 L1(7) -> 6 4.06/1.70 N1(4, 6) -> 3 4.06/1.70 max1(3) -> 1 4.06/1.70 s1(2) -> 7 4.06/1.70 max1(3) -> 7 4.06/1.70 L1(2) -> 5 4.06/1.70 0 -> 1 4.06/1.70 0 -> 7 4.06/1.70 0 -> 2 4.06/1.70 7 -> 1 4.06/1.70 7 -> 2 4.06/1.70 2 -> 1 4.06/1.70 2 -> 7 4.06/1.70 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (4) 4.06/1.70 BOUNDS(1, n^1) 4.06/1.70 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 4.06/1.70 Transformed a relative TRS into a decreasing-loop problem. 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (6) 4.06/1.70 Obligation: 4.06/1.70 Analyzing the following TRS for decreasing loops: 4.06/1.70 4.06/1.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.70 4.06/1.70 4.06/1.70 The TRS R consists of the following rules: 4.06/1.70 4.06/1.70 max(L(x)) -> x 4.06/1.70 max(N(L(0), L(y))) -> y 4.06/1.70 max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y)))) 4.06/1.70 max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z))))) 4.06/1.70 4.06/1.70 S is empty. 4.06/1.70 Rewrite Strategy: INNERMOST 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (7) DecreasingLoopProof (LOWER BOUND(ID)) 4.06/1.70 The following loop(s) give(s) rise to the lower bound Omega(n^1): 4.06/1.70 4.06/1.70 The rewrite sequence 4.06/1.70 4.06/1.70 max(N(L(s(x)), L(s(y)))) ->^+ s(max(N(L(x), L(y)))) 4.06/1.70 4.06/1.70 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 4.06/1.70 4.06/1.70 The pumping substitution is [x / s(x), y / s(y)]. 4.06/1.70 4.06/1.70 The result substitution is [ ]. 4.06/1.70 4.06/1.70 4.06/1.70 4.06/1.70 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (8) 4.06/1.70 Complex Obligation (BEST) 4.06/1.70 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (9) 4.06/1.70 Obligation: 4.06/1.70 Proved the lower bound n^1 for the following obligation: 4.06/1.70 4.06/1.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.70 4.06/1.70 4.06/1.70 The TRS R consists of the following rules: 4.06/1.70 4.06/1.70 max(L(x)) -> x 4.06/1.70 max(N(L(0), L(y))) -> y 4.06/1.70 max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y)))) 4.06/1.70 max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z))))) 4.06/1.70 4.06/1.70 S is empty. 4.06/1.70 Rewrite Strategy: INNERMOST 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (10) LowerBoundPropagationProof (FINISHED) 4.06/1.70 Propagated lower bound. 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (11) 4.06/1.70 BOUNDS(n^1, INF) 4.06/1.70 4.06/1.70 ---------------------------------------- 4.06/1.70 4.06/1.70 (12) 4.06/1.70 Obligation: 4.06/1.70 Analyzing the following TRS for decreasing loops: 4.06/1.70 4.06/1.70 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.70 4.06/1.70 4.06/1.70 The TRS R consists of the following rules: 4.06/1.70 4.06/1.70 max(L(x)) -> x 4.06/1.70 max(N(L(0), L(y))) -> y 4.06/1.70 max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y)))) 4.06/1.70 max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z))))) 4.06/1.70 4.06/1.70 S is empty. 4.06/1.70 Rewrite Strategy: INNERMOST 4.06/1.72 EOF