4.74/1.99 WORST_CASE(Omega(n^1), O(n^1)) 4.74/2.00 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 4.74/2.00 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.74/2.00 4.74/2.00 4.74/2.00 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.74/2.00 4.74/2.00 (0) CpxTRS 4.74/2.00 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 4.74/2.00 (2) CpxTRS 4.74/2.00 (3) CpxTrsMatchBoundsTAProof [FINISHED, 182 ms] 4.74/2.00 (4) BOUNDS(1, n^1) 4.74/2.00 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 4.74/2.00 (6) TRS for Loop Detection 4.74/2.00 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 4.74/2.00 (8) BEST 4.74/2.00 (9) proven lower bound 4.74/2.00 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 4.74/2.00 (11) BOUNDS(n^1, INF) 4.74/2.00 (12) TRS for Loop Detection 4.74/2.00 4.74/2.00 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (0) 4.74/2.00 Obligation: 4.74/2.00 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.74/2.00 4.74/2.00 4.74/2.00 The TRS R consists of the following rules: 4.74/2.00 4.74/2.00 h(x, c(y, z)) -> h(c(s(y), x), z) 4.74/2.00 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.74/2.00 4.74/2.00 S is empty. 4.74/2.00 Rewrite Strategy: FULL 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 4.74/2.00 transformed relative TRS to TRS 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (2) 4.74/2.00 Obligation: 4.74/2.00 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 4.74/2.00 4.74/2.00 4.74/2.00 The TRS R consists of the following rules: 4.74/2.00 4.74/2.00 h(x, c(y, z)) -> h(c(s(y), x), z) 4.74/2.00 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.74/2.00 4.74/2.00 S is empty. 4.74/2.00 Rewrite Strategy: FULL 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (3) CpxTrsMatchBoundsTAProof (FINISHED) 4.74/2.00 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. 4.74/2.00 4.74/2.00 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 4.74/2.00 final states : [1] 4.74/2.00 transitions: 4.74/2.00 c0(0, 0) -> 0 4.74/2.00 s0(0) -> 0 4.74/2.00 00() -> 0 4.74/2.00 h0(0, 0) -> 1 4.74/2.00 s1(0) -> 3 4.74/2.00 c1(3, 0) -> 2 4.74/2.00 h1(2, 0) -> 1 4.74/2.00 01() -> 6 4.74/2.00 s1(6) -> 5 4.74/2.00 c1(0, 0) -> 7 4.74/2.00 c1(5, 7) -> 4 4.74/2.00 h1(0, 4) -> 1 4.74/2.00 c1(3, 2) -> 2 4.74/2.00 s2(5) -> 9 4.74/2.00 c2(9, 0) -> 8 4.74/2.00 h2(8, 7) -> 1 4.74/2.00 c1(0, 4) -> 7 4.74/2.00 h1(2, 4) -> 1 4.74/2.00 s2(0) -> 9 4.74/2.00 c2(9, 8) -> 8 4.74/2.00 h2(8, 0) -> 1 4.74/2.00 h2(8, 4) -> 1 4.74/2.00 c1(5, 7) -> 7 4.74/2.00 c2(9, 2) -> 8 4.74/2.00 c1(3, 8) -> 2 4.74/2.00 c1(0, 7) -> 7 4.74/2.00 c1(5, 0) -> 7 4.74/2.00 c1(5, 4) -> 7 4.74/2.00 h1(8, 4) -> 1 4.74/2.00 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (4) 4.74/2.00 BOUNDS(1, n^1) 4.74/2.00 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 4.74/2.00 Transformed a relative TRS into a decreasing-loop problem. 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (6) 4.74/2.00 Obligation: 4.74/2.00 Analyzing the following TRS for decreasing loops: 4.74/2.00 4.74/2.00 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.74/2.00 4.74/2.00 4.74/2.00 The TRS R consists of the following rules: 4.74/2.00 4.74/2.00 h(x, c(y, z)) -> h(c(s(y), x), z) 4.74/2.00 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.74/2.00 4.74/2.00 S is empty. 4.74/2.00 Rewrite Strategy: FULL 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (7) DecreasingLoopProof (LOWER BOUND(ID)) 4.74/2.00 The following loop(s) give(s) rise to the lower bound Omega(n^1): 4.74/2.00 4.74/2.00 The rewrite sequence 4.74/2.00 4.74/2.00 h(c(s(x), c(s(0), y)), z) ->^+ h(y, c(s(0), c(x, z))) 4.74/2.00 4.74/2.00 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 4.74/2.00 4.74/2.00 The pumping substitution is [y / c(s(x), c(s(0), y))]. 4.74/2.00 4.74/2.00 The result substitution is [z / c(s(0), c(x, z))]. 4.74/2.00 4.74/2.00 4.74/2.00 4.74/2.00 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (8) 4.74/2.00 Complex Obligation (BEST) 4.74/2.00 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (9) 4.74/2.00 Obligation: 4.74/2.00 Proved the lower bound n^1 for the following obligation: 4.74/2.00 4.74/2.00 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.74/2.00 4.74/2.00 4.74/2.00 The TRS R consists of the following rules: 4.74/2.00 4.74/2.00 h(x, c(y, z)) -> h(c(s(y), x), z) 4.74/2.00 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.74/2.00 4.74/2.00 S is empty. 4.74/2.00 Rewrite Strategy: FULL 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (10) LowerBoundPropagationProof (FINISHED) 4.74/2.00 Propagated lower bound. 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (11) 4.74/2.00 BOUNDS(n^1, INF) 4.74/2.00 4.74/2.00 ---------------------------------------- 4.74/2.00 4.74/2.00 (12) 4.74/2.00 Obligation: 4.74/2.00 Analyzing the following TRS for decreasing loops: 4.74/2.00 4.74/2.00 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.74/2.00 4.74/2.00 4.74/2.00 The TRS R consists of the following rules: 4.74/2.00 4.74/2.00 h(x, c(y, z)) -> h(c(s(y), x), z) 4.74/2.00 h(c(s(x), c(s(0), y)), z) -> h(y, c(s(0), c(x, z))) 4.74/2.00 4.74/2.00 S is empty. 4.74/2.00 Rewrite Strategy: FULL 4.85/2.07 EOF