3.29/1.61 WORST_CASE(Omega(n^1), O(n^1)) 3.29/1.62 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 3.29/1.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 3.29/1.62 3.29/1.62 3.29/1.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.29/1.62 3.29/1.62 (0) CpxTRS 3.29/1.62 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 3.29/1.62 (2) CpxTRS 3.29/1.62 (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms] 3.29/1.62 (4) BOUNDS(1, n^1) 3.29/1.62 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 3.29/1.62 (6) TRS for Loop Detection 3.29/1.62 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 3.29/1.62 (8) BEST 3.29/1.62 (9) proven lower bound 3.29/1.62 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 3.29/1.62 (11) BOUNDS(n^1, INF) 3.29/1.62 (12) TRS for Loop Detection 3.29/1.62 3.29/1.62 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (0) 3.29/1.62 Obligation: 3.29/1.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.29/1.62 3.29/1.62 3.29/1.62 The TRS R consists of the following rules: 3.29/1.62 3.29/1.62 p(m, n, s(r)) -> p(m, r, n) 3.29/1.62 p(m, s(n), 0) -> p(0, n, m) 3.29/1.62 p(m, 0, 0) -> m 3.29/1.62 3.29/1.62 S is empty. 3.29/1.62 Rewrite Strategy: INNERMOST 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 3.29/1.62 transformed relative TRS to TRS 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (2) 3.29/1.62 Obligation: 3.29/1.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 3.29/1.62 3.29/1.62 3.29/1.62 The TRS R consists of the following rules: 3.29/1.62 3.29/1.62 p(m, n, s(r)) -> p(m, r, n) 3.29/1.62 p(m, s(n), 0) -> p(0, n, m) 3.29/1.62 p(m, 0, 0) -> m 3.29/1.62 3.29/1.62 S is empty. 3.29/1.62 Rewrite Strategy: INNERMOST 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (3) CpxTrsMatchBoundsTAProof (FINISHED) 3.29/1.62 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. 3.29/1.62 3.29/1.62 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 3.29/1.62 final states : [1] 3.29/1.62 transitions: 3.29/1.62 s0(0) -> 0 3.29/1.62 00() -> 0 3.29/1.62 p0(0, 0, 0) -> 1 3.29/1.62 p1(0, 0, 0) -> 1 3.29/1.62 01() -> 2 3.29/1.62 p1(2, 0, 0) -> 1 3.29/1.62 p1(2, 0, 2) -> 1 3.29/1.62 0 -> 1 3.29/1.62 2 -> 1 3.29/1.62 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (4) 3.29/1.62 BOUNDS(1, n^1) 3.29/1.62 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 3.29/1.62 Transformed a relative TRS into a decreasing-loop problem. 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (6) 3.29/1.62 Obligation: 3.29/1.62 Analyzing the following TRS for decreasing loops: 3.29/1.62 3.29/1.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.29/1.62 3.29/1.62 3.29/1.62 The TRS R consists of the following rules: 3.29/1.62 3.29/1.62 p(m, n, s(r)) -> p(m, r, n) 3.29/1.62 p(m, s(n), 0) -> p(0, n, m) 3.29/1.62 p(m, 0, 0) -> m 3.29/1.62 3.29/1.62 S is empty. 3.29/1.62 Rewrite Strategy: INNERMOST 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (7) DecreasingLoopProof (LOWER BOUND(ID)) 3.29/1.62 The following loop(s) give(s) rise to the lower bound Omega(n^1): 3.29/1.62 3.29/1.62 The rewrite sequence 3.29/1.62 3.29/1.62 p(m, s(r3_0), s(r)) ->^+ p(m, r3_0, r) 3.29/1.62 3.29/1.62 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 3.29/1.62 3.29/1.62 The pumping substitution is [r3_0 / s(r3_0), r / s(r)]. 3.29/1.62 3.29/1.62 The result substitution is [ ]. 3.29/1.62 3.29/1.62 3.29/1.62 3.29/1.62 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (8) 3.29/1.62 Complex Obligation (BEST) 3.29/1.62 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (9) 3.29/1.62 Obligation: 3.29/1.62 Proved the lower bound n^1 for the following obligation: 3.29/1.62 3.29/1.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.29/1.62 3.29/1.62 3.29/1.62 The TRS R consists of the following rules: 3.29/1.62 3.29/1.62 p(m, n, s(r)) -> p(m, r, n) 3.29/1.62 p(m, s(n), 0) -> p(0, n, m) 3.29/1.62 p(m, 0, 0) -> m 3.29/1.62 3.29/1.62 S is empty. 3.29/1.62 Rewrite Strategy: INNERMOST 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (10) LowerBoundPropagationProof (FINISHED) 3.29/1.62 Propagated lower bound. 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (11) 3.29/1.62 BOUNDS(n^1, INF) 3.29/1.62 3.29/1.62 ---------------------------------------- 3.29/1.62 3.29/1.62 (12) 3.29/1.62 Obligation: 3.29/1.62 Analyzing the following TRS for decreasing loops: 3.29/1.62 3.29/1.62 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 3.29/1.62 3.29/1.62 3.29/1.62 The TRS R consists of the following rules: 3.29/1.62 3.29/1.62 p(m, n, s(r)) -> p(m, r, n) 3.29/1.62 p(m, s(n), 0) -> p(0, n, m) 3.29/1.62 p(m, 0, 0) -> m 3.29/1.62 3.29/1.62 S is empty. 3.29/1.62 Rewrite Strategy: INNERMOST 3.29/1.64 EOF