/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) CpxTrsMatchBoundsTAProof [FINISHED, 77 ms] (4) BOUNDS(1, n^1) (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: f(X) -> if(X, c, n__f(n__true)) if(true, X, Y) -> X if(false, X, Y) -> activate(Y) f(X) -> n__f(X) true -> n__true activate(n__f(X)) -> f(activate(X)) activate(n__true) -> true activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: f(X) -> if(X, c, n__f(n__true)) if(true, X, Y) -> X if(false, X, Y) -> activate(Y) f(X) -> n__f(X) true -> n__true activate(n__f(X)) -> f(activate(X)) activate(n__true) -> true activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) CpxTrsMatchBoundsTAProof (FINISHED) 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 4. The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: final states : [1, 2, 3, 4] transitions: c0() -> 0 n__f0(0) -> 0 n__true0() -> 0 false0() -> 0 f0(0) -> 1 if0(0, 0, 0) -> 2 true0() -> 3 activate0(0) -> 4 c1() -> 5 n__true1() -> 7 n__f1(7) -> 6 if1(0, 5, 6) -> 1 activate1(0) -> 2 n__f1(0) -> 1 n__true1() -> 3 activate1(0) -> 8 f1(8) -> 4 true1() -> 4 c2() -> 9 n__true2() -> 11 n__f2(11) -> 10 if2(8, 9, 10) -> 4 activate1(6) -> 1 n__f2(8) -> 4 n__true2() -> 4 f1(8) -> 2 f1(8) -> 8 true1() -> 2 true1() -> 8 if2(8, 9, 10) -> 2 if2(8, 9, 10) -> 8 n__f2(8) -> 2 n__f2(8) -> 8 n__true2() -> 2 n__true2() -> 8 activate2(7) -> 12 f2(12) -> 1 activate1(10) -> 4 activate1(10) -> 2 activate1(10) -> 8 activate2(11) -> 12 f2(12) -> 4 c3() -> 13 n__true3() -> 15 n__f3(15) -> 14 if3(12, 13, 14) -> 1 n__f3(12) -> 1 true2() -> 12 f2(12) -> 2 f2(12) -> 8 if3(12, 13, 14) -> 4 n__f3(12) -> 4 true3() -> 12 n__true3() -> 12 if3(12, 13, 14) -> 2 if3(12, 13, 14) -> 8 n__f3(12) -> 2 n__f3(12) -> 8 n__true4() -> 12 0 -> 4 0 -> 2 0 -> 8 6 -> 1 9 -> 4 9 -> 2 9 -> 8 10 -> 4 10 -> 2 10 -> 8 7 -> 12 11 -> 12 13 -> 1 13 -> 4 13 -> 2 13 -> 8 ---------------------------------------- (4) BOUNDS(1, n^1) ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: f(X) -> if(X, c, n__f(n__true)) if(true, X, Y) -> X if(false, X, Y) -> activate(Y) f(X) -> n__f(X) true -> n__true activate(n__f(X)) -> f(activate(X)) activate(n__true) -> true activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence activate(n__f(X)) ->^+ f(activate(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / n__f(X)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: f(X) -> if(X, c, n__f(n__true)) if(true, X, Y) -> X if(false, X, Y) -> activate(Y) f(X) -> n__f(X) true -> n__true activate(n__f(X)) -> f(activate(X)) activate(n__true) -> true activate(X) -> X S is empty. Rewrite Strategy: FULL ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: f(X) -> if(X, c, n__f(n__true)) if(true, X, Y) -> X if(false, X, Y) -> activate(Y) f(X) -> n__f(X) true -> n__true activate(n__f(X)) -> f(activate(X)) activate(n__true) -> true activate(X) -> X S is empty. Rewrite Strategy: FULL