4.13/1.80 WORST_CASE(Omega(n^1), O(n^1)) 4.13/1.80 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 4.13/1.80 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.13/1.80 4.13/1.80 4.13/1.80 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.13/1.80 4.13/1.80 (0) CpxTRS 4.13/1.80 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 4.13/1.80 (2) CpxTRS 4.13/1.80 (3) CpxTrsMatchBoundsTAProof [FINISHED, 116 ms] 4.13/1.80 (4) BOUNDS(1, n^1) 4.13/1.80 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 4.13/1.80 (6) TRS for Loop Detection 4.13/1.80 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 4.13/1.80 (8) BEST 4.13/1.80 (9) proven lower bound 4.13/1.80 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 4.13/1.80 (11) BOUNDS(n^1, INF) 4.13/1.80 (12) TRS for Loop Detection 4.13/1.80 4.13/1.80 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (0) 4.13/1.80 Obligation: 4.13/1.80 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.13/1.80 4.13/1.80 4.13/1.80 The TRS R consists of the following rules: 4.13/1.80 4.13/1.80 h(x, c(y, z)) -> h(c(s(y), x), z) 4.13/1.80 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.13/1.80 4.13/1.80 S is empty. 4.13/1.80 Rewrite Strategy: INNERMOST 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 4.13/1.80 transformed relative TRS to TRS 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (2) 4.13/1.80 Obligation: 4.13/1.80 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 4.13/1.80 4.13/1.80 4.13/1.80 The TRS R consists of the following rules: 4.13/1.80 4.13/1.80 h(x, c(y, z)) -> h(c(s(y), x), z) 4.13/1.80 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.13/1.80 4.13/1.80 S is empty. 4.13/1.80 Rewrite Strategy: INNERMOST 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (3) CpxTrsMatchBoundsTAProof (FINISHED) 4.13/1.80 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 2. 4.13/1.80 4.13/1.80 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 4.13/1.80 final states : [1] 4.13/1.80 transitions: 4.13/1.80 c0(0, 0) -> 0 4.13/1.80 s0(0) -> 0 4.13/1.80 00() -> 0 4.13/1.80 h0(0, 0) -> 1 4.13/1.80 s1(0) -> 3 4.13/1.80 c1(3, 0) -> 2 4.13/1.80 h1(2, 0) -> 1 4.13/1.80 01() -> 6 4.13/1.80 s1(6) -> 5 4.13/1.80 c1(0, 0) -> 7 4.13/1.80 c1(5, 7) -> 4 4.13/1.80 h1(0, 4) -> 1 4.13/1.80 c1(3, 2) -> 2 4.13/1.80 s2(5) -> 9 4.13/1.80 c2(9, 0) -> 8 4.13/1.80 h2(8, 7) -> 1 4.13/1.80 c1(0, 4) -> 7 4.13/1.80 h1(2, 4) -> 1 4.13/1.80 s2(0) -> 9 4.13/1.80 c2(9, 8) -> 8 4.13/1.80 h2(8, 0) -> 1 4.13/1.80 h2(8, 4) -> 1 4.13/1.80 c1(5, 7) -> 7 4.13/1.80 c2(9, 2) -> 8 4.13/1.80 c1(3, 8) -> 2 4.13/1.80 c1(0, 7) -> 7 4.13/1.80 c1(5, 0) -> 7 4.13/1.80 c1(5, 4) -> 7 4.13/1.80 h1(8, 4) -> 1 4.13/1.80 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (4) 4.13/1.80 BOUNDS(1, n^1) 4.13/1.80 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 4.13/1.80 Transformed a relative TRS into a decreasing-loop problem. 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (6) 4.13/1.80 Obligation: 4.13/1.80 Analyzing the following TRS for decreasing loops: 4.13/1.80 4.13/1.80 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.13/1.80 4.13/1.80 4.13/1.80 The TRS R consists of the following rules: 4.13/1.80 4.13/1.80 h(x, c(y, z)) -> h(c(s(y), x), z) 4.13/1.80 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.13/1.80 4.13/1.80 S is empty. 4.13/1.80 Rewrite Strategy: INNERMOST 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (7) DecreasingLoopProof (LOWER BOUND(ID)) 4.13/1.80 The following loop(s) give(s) rise to the lower bound Omega(n^1): 4.13/1.80 4.13/1.80 The rewrite sequence 4.13/1.80 4.13/1.80 h(c(s(x), c(s(0), y)), z) ->^+ h(y, c(s(0), c(x, z))) 4.13/1.80 4.13/1.80 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 4.13/1.80 4.13/1.80 The pumping substitution is [y / c(s(x), c(s(0), y))]. 4.13/1.80 4.13/1.80 The result substitution is [z / c(s(0), c(x, z))]. 4.13/1.80 4.13/1.80 4.13/1.80 4.13/1.80 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (8) 4.13/1.80 Complex Obligation (BEST) 4.13/1.80 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (9) 4.13/1.80 Obligation: 4.13/1.80 Proved the lower bound n^1 for the following obligation: 4.13/1.80 4.13/1.80 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.13/1.80 4.13/1.80 4.13/1.80 The TRS R consists of the following rules: 4.13/1.80 4.13/1.80 h(x, c(y, z)) -> h(c(s(y), x), z) 4.13/1.80 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.13/1.80 4.13/1.80 S is empty. 4.13/1.80 Rewrite Strategy: INNERMOST 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (10) LowerBoundPropagationProof (FINISHED) 4.13/1.80 Propagated lower bound. 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (11) 4.13/1.80 BOUNDS(n^1, INF) 4.13/1.80 4.13/1.80 ---------------------------------------- 4.13/1.80 4.13/1.80 (12) 4.13/1.80 Obligation: 4.13/1.80 Analyzing the following TRS for decreasing loops: 4.13/1.80 4.13/1.80 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.13/1.80 4.13/1.80 4.13/1.80 The TRS R consists of the following rules: 4.13/1.80 4.13/1.80 h(x, c(y, z)) -> h(c(s(y), x), z) 4.13/1.80 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.13/1.80 4.13/1.80 S is empty. 4.13/1.80 Rewrite Strategy: INNERMOST 4.29/7.46 EOF