4.02/1.72 WORST_CASE(Omega(n^1), O(n^1)) 4.06/1.74 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 4.06/1.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 4.06/1.74 4.06/1.74 4.06/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.74 4.06/1.74 (0) CpxTRS 4.06/1.74 (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 4.06/1.74 (2) CpxTRS 4.06/1.74 (3) CpxTrsMatchBoundsTAProof [FINISHED, 116 ms] 4.06/1.74 (4) BOUNDS(1, n^1) 4.06/1.74 (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 4.06/1.74 (6) TRS for Loop Detection 4.06/1.74 (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 4.06/1.74 (8) BEST 4.06/1.74 (9) proven lower bound 4.06/1.74 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 4.06/1.74 (11) BOUNDS(n^1, INF) 4.06/1.74 (12) TRS for Loop Detection 4.06/1.74 4.06/1.74 4.06/1.74 ---------------------------------------- 4.06/1.74 4.06/1.74 (0) 4.06/1.74 Obligation: 4.06/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.74 4.06/1.74 4.06/1.74 The TRS R consists of the following rules: 4.06/1.74 4.06/1.74 not(true) -> false 4.06/1.74 not(false) -> true 4.06/1.74 evenodd(x, 0) -> not(evenodd(x, s(0))) 4.06/1.74 evenodd(0, s(0)) -> false 4.06/1.74 evenodd(s(x), s(0)) -> evenodd(x, 0) 4.06/1.74 4.06/1.74 S is empty. 4.06/1.74 Rewrite Strategy: FULL 4.06/1.74 ---------------------------------------- 4.06/1.74 4.06/1.74 (1) RelTrsToTrsProof (UPPER BOUND(ID)) 4.06/1.74 transformed relative TRS to TRS 4.06/1.74 ---------------------------------------- 4.06/1.74 4.06/1.74 (2) 4.06/1.74 Obligation: 4.06/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 4.06/1.74 4.06/1.74 4.06/1.74 The TRS R consists of the following rules: 4.06/1.74 4.06/1.74 not(true) -> false 4.06/1.74 not(false) -> true 4.06/1.74 evenodd(x, 0) -> not(evenodd(x, s(0))) 4.06/1.74 evenodd(0, s(0)) -> false 4.06/1.74 evenodd(s(x), s(0)) -> evenodd(x, 0) 4.06/1.74 4.06/1.74 S is empty. 4.06/1.74 Rewrite Strategy: FULL 4.06/1.74 ---------------------------------------- 4.06/1.74 4.06/1.74 (3) CpxTrsMatchBoundsTAProof (FINISHED) 4.06/1.74 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 3. 4.06/1.74 4.06/1.74 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 4.06/1.74 final states : [1, 2] 4.06/1.74 transitions: 4.06/1.74 true0() -> 0 4.06/1.74 false0() -> 0 4.06/1.74 00() -> 0 4.06/1.74 s0(0) -> 0 4.06/1.74 not0(0) -> 1 4.06/1.74 evenodd0(0, 0) -> 2 4.06/1.74 false1() -> 1 4.06/1.74 true1() -> 1 4.06/1.74 01() -> 5 4.06/1.74 s1(5) -> 4 4.06/1.74 evenodd1(0, 4) -> 3 4.06/1.74 not1(3) -> 2 4.06/1.74 false1() -> 2 4.06/1.74 01() -> 6 4.06/1.74 evenodd1(0, 6) -> 2 4.06/1.74 02() -> 9 4.06/1.74 s2(9) -> 8 4.06/1.74 evenodd2(0, 8) -> 7 4.06/1.74 not2(7) -> 2 4.06/1.74 false1() -> 3 4.06/1.74 evenodd1(0, 6) -> 3 4.06/1.74 true2() -> 2 4.06/1.74 not2(7) -> 3 4.06/1.74 false1() -> 7 4.06/1.74 evenodd1(0, 6) -> 7 4.06/1.74 not2(7) -> 7 4.06/1.74 true2() -> 3 4.06/1.74 true2() -> 7 4.06/1.74 false2() -> 2 4.06/1.74 false3() -> 2 4.06/1.74 false3() -> 3 4.06/1.74 false3() -> 7 4.06/1.74 true3() -> 2 4.06/1.74 true3() -> 3 4.06/1.74 true3() -> 7 4.06/1.74 4.06/1.74 ---------------------------------------- 4.06/1.74 4.06/1.74 (4) 4.06/1.74 BOUNDS(1, n^1) 4.06/1.74 4.06/1.74 ---------------------------------------- 4.06/1.74 4.06/1.74 (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 4.06/1.74 Transformed a relative TRS into a decreasing-loop problem. 4.06/1.74 ---------------------------------------- 4.06/1.74 4.06/1.74 (6) 4.06/1.74 Obligation: 4.06/1.74 Analyzing the following TRS for decreasing loops: 4.06/1.74 4.06/1.74 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.74 4.06/1.74 4.06/1.74 The TRS R consists of the following rules: 4.06/1.74 4.06/1.74 not(true) -> false 4.06/1.74 not(false) -> true 4.06/1.74 evenodd(x, 0) -> not(evenodd(x, s(0))) 4.06/1.74 evenodd(0, s(0)) -> false 4.06/1.74 evenodd(s(x), s(0)) -> evenodd(x, 0) 4.06/1.75 4.06/1.75 S is empty. 4.06/1.75 Rewrite Strategy: FULL 4.06/1.75 ---------------------------------------- 4.06/1.75 4.06/1.75 (7) DecreasingLoopProof (LOWER BOUND(ID)) 4.06/1.75 The following loop(s) give(s) rise to the lower bound Omega(n^1): 4.06/1.75 4.06/1.75 The rewrite sequence 4.06/1.75 4.06/1.75 evenodd(s(x1_0), 0) ->^+ not(evenodd(x1_0, 0)) 4.06/1.75 4.06/1.75 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 4.06/1.75 4.06/1.75 The pumping substitution is [x1_0 / s(x1_0)]. 4.06/1.75 4.06/1.75 The result substitution is [ ]. 4.06/1.75 4.06/1.75 4.06/1.75 4.06/1.75 4.06/1.75 ---------------------------------------- 4.06/1.75 4.06/1.75 (8) 4.06/1.75 Complex Obligation (BEST) 4.06/1.75 4.06/1.75 ---------------------------------------- 4.06/1.75 4.06/1.75 (9) 4.06/1.75 Obligation: 4.06/1.75 Proved the lower bound n^1 for the following obligation: 4.06/1.75 4.06/1.75 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.75 4.06/1.75 4.06/1.75 The TRS R consists of the following rules: 4.06/1.75 4.06/1.75 not(true) -> false 4.06/1.75 not(false) -> true 4.06/1.75 evenodd(x, 0) -> not(evenodd(x, s(0))) 4.06/1.75 evenodd(0, s(0)) -> false 4.06/1.75 evenodd(s(x), s(0)) -> evenodd(x, 0) 4.06/1.75 4.06/1.75 S is empty. 4.06/1.75 Rewrite Strategy: FULL 4.06/1.75 ---------------------------------------- 4.06/1.75 4.06/1.75 (10) LowerBoundPropagationProof (FINISHED) 4.06/1.75 Propagated lower bound. 4.06/1.75 ---------------------------------------- 4.06/1.75 4.06/1.75 (11) 4.06/1.75 BOUNDS(n^1, INF) 4.06/1.75 4.06/1.75 ---------------------------------------- 4.06/1.75 4.06/1.75 (12) 4.06/1.75 Obligation: 4.06/1.75 Analyzing the following TRS for decreasing loops: 4.06/1.75 4.06/1.75 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 4.06/1.75 4.06/1.75 4.06/1.75 The TRS R consists of the following rules: 4.06/1.75 4.06/1.75 not(true) -> false 4.06/1.75 not(false) -> true 4.06/1.75 evenodd(x, 0) -> not(evenodd(x, s(0))) 4.06/1.75 evenodd(0, s(0)) -> false 4.06/1.75 evenodd(s(x), s(0)) -> evenodd(x, 0) 4.06/1.75 4.06/1.75 S is empty. 4.06/1.75 Rewrite Strategy: FULL 4.06/1.78 EOF