29.46/11.67 WORST_CASE(Omega(n^1), O(n^1)) 29.46/11.67 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 29.46/11.67 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 29.46/11.67 29.46/11.67 29.46/11.67 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.46/11.67 29.46/11.67 (0) CpxTRS 29.46/11.67 (1) DependencyGraphProof [UPPER BOUND(ID), 0 ms] 29.46/11.67 (2) CpxTRS 29.46/11.67 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 29.46/11.67 (4) CpxTRS 29.46/11.67 (5) CpxTrsMatchBoundsTAProof [FINISHED, 290 ms] 29.46/11.67 (6) BOUNDS(1, n^1) 29.46/11.67 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 29.46/11.67 (8) TRS for Loop Detection 29.46/11.67 (9) DecreasingLoopProof [LOWER BOUND(ID), 63 ms] 29.46/11.67 (10) BEST 29.46/11.67 (11) proven lower bound 29.46/11.67 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 29.46/11.67 (13) BOUNDS(n^1, INF) 29.46/11.67 (14) TRS for Loop Detection 29.46/11.67 29.46/11.67 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (0) 29.46/11.67 Obligation: 29.46/11.67 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.46/11.67 29.46/11.67 29.46/11.67 The TRS R consists of the following rules: 29.46/11.67 29.46/11.67 minus(n__0, Y) -> 0 29.46/11.67 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) 29.46/11.67 geq(X, n__0) -> true 29.46/11.67 geq(n__0, n__s(Y)) -> false 29.46/11.67 geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) 29.46/11.67 div(0, n__s(Y)) -> 0 29.46/11.67 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0) 29.46/11.67 if(true, X, Y) -> activate(X) 29.46/11.67 if(false, X, Y) -> activate(Y) 29.46/11.67 0 -> n__0 29.46/11.67 s(X) -> n__s(X) 29.46/11.67 activate(n__0) -> 0 29.46/11.67 activate(n__s(X)) -> s(X) 29.46/11.67 activate(X) -> X 29.46/11.67 29.46/11.67 S is empty. 29.46/11.67 Rewrite Strategy: FULL 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (1) DependencyGraphProof (UPPER BOUND(ID)) 29.46/11.67 The following rules are not reachable from basic terms in the dependency graph and can be removed: 29.46/11.67 29.46/11.67 div(0, n__s(Y)) -> 0 29.46/11.67 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0) 29.46/11.67 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (2) 29.46/11.67 Obligation: 29.46/11.67 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 29.46/11.67 29.46/11.67 29.46/11.67 The TRS R consists of the following rules: 29.46/11.67 29.46/11.67 minus(n__0, Y) -> 0 29.46/11.67 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) 29.46/11.67 geq(X, n__0) -> true 29.46/11.67 geq(n__0, n__s(Y)) -> false 29.46/11.67 geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) 29.46/11.67 if(true, X, Y) -> activate(X) 29.46/11.67 if(false, X, Y) -> activate(Y) 29.46/11.67 0 -> n__0 29.46/11.67 s(X) -> n__s(X) 29.46/11.67 activate(n__0) -> 0 29.46/11.67 activate(n__s(X)) -> s(X) 29.46/11.67 activate(X) -> X 29.46/11.67 29.46/11.67 S is empty. 29.46/11.67 Rewrite Strategy: FULL 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 29.46/11.67 transformed relative TRS to TRS 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (4) 29.46/11.67 Obligation: 29.46/11.67 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 29.46/11.67 29.46/11.67 29.46/11.67 The TRS R consists of the following rules: 29.46/11.67 29.46/11.67 minus(n__0, Y) -> 0 29.46/11.67 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) 29.46/11.67 geq(X, n__0) -> true 29.46/11.67 geq(n__0, n__s(Y)) -> false 29.46/11.67 geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) 29.46/11.67 if(true, X, Y) -> activate(X) 29.46/11.67 if(false, X, Y) -> activate(Y) 29.46/11.67 0 -> n__0 29.46/11.67 s(X) -> n__s(X) 29.46/11.67 activate(n__0) -> 0 29.46/11.67 activate(n__s(X)) -> s(X) 29.46/11.67 activate(X) -> X 29.46/11.67 29.46/11.67 S is empty. 29.46/11.67 Rewrite Strategy: FULL 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (5) CpxTrsMatchBoundsTAProof (FINISHED) 29.46/11.67 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. 29.46/11.67 29.46/11.67 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 29.46/11.67 final states : [1, 2, 3, 4, 5, 6] 29.46/11.67 transitions: 29.46/11.67 n__00() -> 0 29.46/11.67 n__s0(0) -> 0 29.46/11.67 true0() -> 0 29.46/11.67 false0() -> 0 29.46/11.67 minus0(0, 0) -> 1 29.46/11.67 geq0(0, 0) -> 2 29.46/11.67 if0(0, 0, 0) -> 3 29.46/11.67 00() -> 4 29.46/11.67 s0(0) -> 5 29.46/11.67 activate0(0) -> 6 29.46/11.67 01() -> 1 29.46/11.67 activate1(0) -> 7 29.46/11.67 activate1(0) -> 8 29.46/11.67 minus1(7, 8) -> 1 29.46/11.67 true1() -> 2 29.46/11.67 false1() -> 2 29.46/11.67 activate1(0) -> 9 29.46/11.67 activate1(0) -> 10 29.46/11.67 geq1(9, 10) -> 2 29.46/11.67 activate1(0) -> 3 29.46/11.67 n__01() -> 4 29.46/11.67 n__s1(0) -> 5 29.46/11.67 01() -> 6 29.46/11.67 s1(0) -> 6 29.46/11.67 n__02() -> 1 29.46/11.67 n__02() -> 6 29.46/11.67 n__s2(0) -> 6 29.46/11.67 01() -> 3 29.46/11.67 01() -> 7 29.46/11.67 01() -> 8 29.46/11.67 01() -> 9 29.46/11.67 01() -> 10 29.46/11.67 s1(0) -> 3 29.46/11.67 s1(0) -> 7 29.46/11.67 s1(0) -> 8 29.46/11.67 s1(0) -> 9 29.46/11.67 s1(0) -> 10 29.46/11.67 n__02() -> 3 29.46/11.67 n__02() -> 7 29.46/11.67 n__02() -> 8 29.46/11.67 n__02() -> 9 29.46/11.67 n__02() -> 10 29.46/11.67 n__s2(0) -> 3 29.46/11.67 n__s2(0) -> 7 29.46/11.67 n__s2(0) -> 8 29.46/11.67 n__s2(0) -> 9 29.46/11.67 n__s2(0) -> 10 29.46/11.67 02() -> 1 29.46/11.67 activate2(0) -> 11 29.46/11.67 activate2(0) -> 12 29.46/11.67 minus2(11, 12) -> 1 29.46/11.67 true2() -> 2 29.46/11.67 false2() -> 2 29.46/11.67 activate2(0) -> 13 29.46/11.67 activate2(0) -> 14 29.46/11.67 geq2(13, 14) -> 2 29.46/11.67 n__03() -> 1 29.46/11.67 01() -> 11 29.46/11.67 01() -> 12 29.46/11.67 01() -> 13 29.46/11.67 01() -> 14 29.46/11.67 s1(0) -> 11 29.46/11.67 s1(0) -> 12 29.46/11.67 s1(0) -> 13 29.46/11.67 s1(0) -> 14 29.46/11.67 n__02() -> 11 29.46/11.67 n__02() -> 12 29.46/11.67 n__02() -> 13 29.46/11.67 n__02() -> 14 29.46/11.67 n__s2(0) -> 11 29.46/11.67 n__s2(0) -> 12 29.46/11.67 n__s2(0) -> 13 29.46/11.67 n__s2(0) -> 14 29.46/11.67 03() -> 1 29.46/11.67 activate3(0) -> 15 29.46/11.67 activate3(0) -> 16 29.46/11.67 minus3(15, 16) -> 1 29.46/11.67 true3() -> 2 29.46/11.67 false3() -> 2 29.46/11.67 activate3(0) -> 17 29.46/11.67 activate3(0) -> 18 29.46/11.67 geq3(17, 18) -> 2 29.46/11.67 n__04() -> 1 29.46/11.67 01() -> 15 29.46/11.67 01() -> 16 29.46/11.67 01() -> 17 29.46/11.67 01() -> 18 29.46/11.67 s1(0) -> 15 29.46/11.67 s1(0) -> 16 29.46/11.67 s1(0) -> 17 29.46/11.67 s1(0) -> 18 29.46/11.67 n__02() -> 15 29.46/11.67 n__02() -> 16 29.46/11.67 n__02() -> 17 29.46/11.67 n__02() -> 18 29.46/11.67 n__s2(0) -> 15 29.46/11.67 n__s2(0) -> 16 29.46/11.67 n__s2(0) -> 17 29.46/11.67 n__s2(0) -> 18 29.46/11.67 0 -> 6 29.46/11.67 0 -> 3 29.46/11.67 0 -> 7 29.46/11.67 0 -> 8 29.46/11.67 0 -> 9 29.46/11.67 0 -> 10 29.46/11.67 0 -> 11 29.46/11.67 0 -> 12 29.46/11.67 0 -> 13 29.46/11.67 0 -> 14 29.46/11.67 0 -> 15 29.46/11.67 0 -> 16 29.46/11.67 0 -> 17 29.46/11.67 0 -> 18 29.46/11.67 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (6) 29.46/11.67 BOUNDS(1, n^1) 29.46/11.67 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 29.46/11.67 Transformed a relative TRS into a decreasing-loop problem. 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (8) 29.46/11.67 Obligation: 29.46/11.67 Analyzing the following TRS for decreasing loops: 29.46/11.67 29.46/11.67 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.46/11.67 29.46/11.67 29.46/11.67 The TRS R consists of the following rules: 29.46/11.67 29.46/11.67 minus(n__0, Y) -> 0 29.46/11.67 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) 29.46/11.67 geq(X, n__0) -> true 29.46/11.67 geq(n__0, n__s(Y)) -> false 29.46/11.67 geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) 29.46/11.67 div(0, n__s(Y)) -> 0 29.46/11.67 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0) 29.46/11.67 if(true, X, Y) -> activate(X) 29.46/11.67 if(false, X, Y) -> activate(Y) 29.46/11.67 0 -> n__0 29.46/11.67 s(X) -> n__s(X) 29.46/11.67 activate(n__0) -> 0 29.46/11.67 activate(n__s(X)) -> s(X) 29.46/11.67 activate(X) -> X 29.46/11.67 29.46/11.67 S is empty. 29.46/11.67 Rewrite Strategy: FULL 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (9) DecreasingLoopProof (LOWER BOUND(ID)) 29.46/11.67 The following loop(s) give(s) rise to the lower bound Omega(n^1): 29.46/11.67 29.46/11.67 The rewrite sequence 29.46/11.67 29.46/11.67 minus(n__s(X), n__s(Y)) ->^+ minus(X, Y) 29.46/11.67 29.46/11.67 gives rise to a decreasing loop by considering the right hand sides subterm at position []. 29.46/11.67 29.46/11.67 The pumping substitution is [X / n__s(X), Y / n__s(Y)]. 29.46/11.67 29.46/11.67 The result substitution is [ ]. 29.46/11.67 29.46/11.67 29.46/11.67 29.46/11.67 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (10) 29.46/11.67 Complex Obligation (BEST) 29.46/11.67 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (11) 29.46/11.67 Obligation: 29.46/11.67 Proved the lower bound n^1 for the following obligation: 29.46/11.67 29.46/11.67 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.46/11.67 29.46/11.67 29.46/11.67 The TRS R consists of the following rules: 29.46/11.67 29.46/11.67 minus(n__0, Y) -> 0 29.46/11.67 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) 29.46/11.67 geq(X, n__0) -> true 29.46/11.67 geq(n__0, n__s(Y)) -> false 29.46/11.67 geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) 29.46/11.67 div(0, n__s(Y)) -> 0 29.46/11.67 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0) 29.46/11.67 if(true, X, Y) -> activate(X) 29.46/11.67 if(false, X, Y) -> activate(Y) 29.46/11.67 0 -> n__0 29.46/11.67 s(X) -> n__s(X) 29.46/11.67 activate(n__0) -> 0 29.46/11.67 activate(n__s(X)) -> s(X) 29.46/11.67 activate(X) -> X 29.46/11.67 29.46/11.67 S is empty. 29.46/11.67 Rewrite Strategy: FULL 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (12) LowerBoundPropagationProof (FINISHED) 29.46/11.67 Propagated lower bound. 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (13) 29.46/11.67 BOUNDS(n^1, INF) 29.46/11.67 29.46/11.67 ---------------------------------------- 29.46/11.67 29.46/11.67 (14) 29.46/11.67 Obligation: 29.46/11.67 Analyzing the following TRS for decreasing loops: 29.46/11.67 29.46/11.67 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 29.46/11.67 29.46/11.67 29.46/11.67 The TRS R consists of the following rules: 29.46/11.67 29.46/11.67 minus(n__0, Y) -> 0 29.46/11.67 minus(n__s(X), n__s(Y)) -> minus(activate(X), activate(Y)) 29.46/11.67 geq(X, n__0) -> true 29.46/11.67 geq(n__0, n__s(Y)) -> false 29.46/11.67 geq(n__s(X), n__s(Y)) -> geq(activate(X), activate(Y)) 29.46/11.67 div(0, n__s(Y)) -> 0 29.46/11.67 div(s(X), n__s(Y)) -> if(geq(X, activate(Y)), n__s(div(minus(X, activate(Y)), n__s(activate(Y)))), n__0) 29.46/11.67 if(true, X, Y) -> activate(X) 29.46/11.67 if(false, X, Y) -> activate(Y) 29.46/11.67 0 -> n__0 29.46/11.67 s(X) -> n__s(X) 29.46/11.67 activate(n__0) -> 0 29.46/11.67 activate(n__s(X)) -> s(X) 29.46/11.68 activate(X) -> X 29.46/11.68 29.46/11.68 S is empty. 29.46/11.68 Rewrite Strategy: FULL 29.55/11.72 EOF