1096.78/291.51 WORST_CASE(Omega(n^1), O(n^1)) 1097.04/291.53 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1097.04/291.53 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1097.04/291.53 1097.04/291.53 1097.04/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1097.04/291.53 1097.04/291.53 (0) CpxTRS 1097.04/291.53 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 1097.04/291.53 (2) CpxTRS 1097.04/291.53 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 1097.04/291.53 (4) CpxTRS 1097.04/291.53 (5) CpxTrsMatchBoundsTAProof [FINISHED, 369 ms] 1097.04/291.53 (6) BOUNDS(1, n^1) 1097.04/291.53 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1097.04/291.53 (8) TRS for Loop Detection 1097.04/291.53 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1097.04/291.53 (10) BEST 1097.04/291.53 (11) proven lower bound 1097.04/291.53 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 1097.04/291.53 (13) BOUNDS(n^1, INF) 1097.04/291.53 (14) TRS for Loop Detection 1097.04/291.53 1097.04/291.53 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (0) 1097.04/291.53 Obligation: 1097.04/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1097.04/291.53 1097.04/291.53 1097.04/291.53 The TRS R consists of the following rules: 1097.04/291.53 1097.04/291.53 active(zeros) -> mark(cons(0, zeros)) 1097.04/291.53 active(tail(cons(X, XS))) -> mark(XS) 1097.04/291.53 active(cons(X1, X2)) -> cons(active(X1), X2) 1097.04/291.53 active(tail(X)) -> tail(active(X)) 1097.04/291.53 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1097.04/291.53 tail(mark(X)) -> mark(tail(X)) 1097.04/291.53 proper(zeros) -> ok(zeros) 1097.04/291.53 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1097.04/291.53 proper(0) -> ok(0) 1097.04/291.53 proper(tail(X)) -> tail(proper(X)) 1097.04/291.53 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1097.04/291.53 tail(ok(X)) -> ok(tail(X)) 1097.04/291.53 top(mark(X)) -> top(proper(X)) 1097.04/291.53 top(ok(X)) -> top(active(X)) 1097.04/291.53 1097.04/291.53 S is empty. 1097.04/291.53 Rewrite Strategy: FULL 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 1097.04/291.53 The following defined symbols can occur below the 0th argument of cons: active, proper, cons 1097.04/291.53 The following defined symbols can occur below the 1th argument of cons: active, proper, cons 1097.04/291.53 The following defined symbols can occur below the 0th argument of top: active, proper, cons 1097.04/291.53 The following defined symbols can occur below the 0th argument of proper: active, proper, cons 1097.04/291.53 The following defined symbols can occur below the 0th argument of active: active, proper, cons 1097.04/291.53 1097.04/291.53 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 1097.04/291.53 active(tail(cons(X, XS))) -> mark(XS) 1097.04/291.53 active(tail(X)) -> tail(active(X)) 1097.04/291.53 proper(tail(X)) -> tail(proper(X)) 1097.04/291.53 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (2) 1097.04/291.53 Obligation: 1097.04/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 1097.04/291.53 1097.04/291.53 1097.04/291.53 The TRS R consists of the following rules: 1097.04/291.53 1097.04/291.53 active(zeros) -> mark(cons(0, zeros)) 1097.04/291.53 active(cons(X1, X2)) -> cons(active(X1), X2) 1097.04/291.53 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1097.04/291.53 tail(mark(X)) -> mark(tail(X)) 1097.04/291.53 proper(zeros) -> ok(zeros) 1097.04/291.53 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1097.04/291.53 proper(0) -> ok(0) 1097.04/291.53 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1097.04/291.53 tail(ok(X)) -> ok(tail(X)) 1097.04/291.53 top(mark(X)) -> top(proper(X)) 1097.04/291.53 top(ok(X)) -> top(active(X)) 1097.04/291.53 1097.04/291.53 S is empty. 1097.04/291.53 Rewrite Strategy: FULL 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 1097.04/291.53 transformed relative TRS to TRS 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (4) 1097.04/291.53 Obligation: 1097.04/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 1097.04/291.53 1097.04/291.53 1097.04/291.53 The TRS R consists of the following rules: 1097.04/291.53 1097.04/291.53 active(zeros) -> mark(cons(0, zeros)) 1097.04/291.53 active(cons(X1, X2)) -> cons(active(X1), X2) 1097.04/291.53 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1097.04/291.53 tail(mark(X)) -> mark(tail(X)) 1097.04/291.53 proper(zeros) -> ok(zeros) 1097.04/291.53 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1097.04/291.53 proper(0) -> ok(0) 1097.04/291.53 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1097.04/291.53 tail(ok(X)) -> ok(tail(X)) 1097.04/291.53 top(mark(X)) -> top(proper(X)) 1097.04/291.53 top(ok(X)) -> top(active(X)) 1097.04/291.53 1097.04/291.53 S is empty. 1097.04/291.53 Rewrite Strategy: FULL 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (5) CpxTrsMatchBoundsTAProof (FINISHED) 1097.04/291.53 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 5. 1097.04/291.53 1097.04/291.53 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 1097.04/291.53 final states : [1, 2, 3, 4, 5] 1097.04/291.53 transitions: 1097.04/291.53 zeros0() -> 0 1097.04/291.53 mark0(0) -> 0 1097.04/291.53 00() -> 0 1097.04/291.53 ok0(0) -> 0 1097.04/291.53 active0(0) -> 1 1097.04/291.53 cons0(0, 0) -> 2 1097.04/291.53 tail0(0) -> 3 1097.04/291.53 proper0(0) -> 4 1097.04/291.53 top0(0) -> 5 1097.04/291.53 01() -> 7 1097.04/291.53 zeros1() -> 8 1097.04/291.53 cons1(7, 8) -> 6 1097.04/291.53 mark1(6) -> 1 1097.04/291.53 cons1(0, 0) -> 9 1097.04/291.53 mark1(9) -> 2 1097.04/291.53 tail1(0) -> 10 1097.04/291.53 mark1(10) -> 3 1097.04/291.53 zeros1() -> 11 1097.04/291.53 ok1(11) -> 4 1097.04/291.53 01() -> 12 1097.04/291.53 ok1(12) -> 4 1097.04/291.53 cons1(0, 0) -> 13 1097.04/291.53 ok1(13) -> 2 1097.04/291.53 tail1(0) -> 14 1097.04/291.53 ok1(14) -> 3 1097.04/291.53 proper1(0) -> 15 1097.04/291.53 top1(15) -> 5 1097.04/291.53 active1(0) -> 16 1097.04/291.53 top1(16) -> 5 1097.04/291.53 mark1(6) -> 16 1097.04/291.53 mark1(9) -> 9 1097.04/291.53 mark1(9) -> 13 1097.04/291.53 mark1(10) -> 10 1097.04/291.53 mark1(10) -> 14 1097.04/291.53 ok1(11) -> 15 1097.04/291.53 ok1(12) -> 15 1097.04/291.53 ok1(13) -> 9 1097.04/291.53 ok1(13) -> 13 1097.04/291.53 ok1(14) -> 10 1097.04/291.53 ok1(14) -> 14 1097.04/291.53 proper2(6) -> 17 1097.04/291.53 top2(17) -> 5 1097.04/291.53 active2(11) -> 18 1097.04/291.53 top2(18) -> 5 1097.04/291.53 active2(12) -> 18 1097.04/291.53 02() -> 20 1097.04/291.53 zeros2() -> 21 1097.04/291.53 cons2(20, 21) -> 19 1097.04/291.53 mark2(19) -> 18 1097.04/291.53 proper2(7) -> 22 1097.04/291.53 proper2(8) -> 23 1097.04/291.53 cons2(22, 23) -> 17 1097.04/291.53 zeros2() -> 24 1097.04/291.53 ok2(24) -> 23 1097.04/291.53 02() -> 25 1097.04/291.53 ok2(25) -> 22 1097.04/291.53 proper3(19) -> 26 1097.04/291.53 top3(26) -> 5 1097.04/291.53 proper3(20) -> 27 1097.04/291.53 proper3(21) -> 28 1097.04/291.53 cons3(27, 28) -> 26 1097.04/291.53 cons3(25, 24) -> 29 1097.04/291.53 ok3(29) -> 17 1097.04/291.53 zeros3() -> 30 1097.04/291.53 ok3(30) -> 28 1097.04/291.53 03() -> 31 1097.04/291.53 ok3(31) -> 27 1097.04/291.53 active3(29) -> 32 1097.04/291.53 top3(32) -> 5 1097.04/291.53 cons4(31, 30) -> 33 1097.04/291.53 ok4(33) -> 26 1097.04/291.53 active4(25) -> 34 1097.04/291.53 cons4(34, 24) -> 32 1097.04/291.53 active4(33) -> 35 1097.04/291.53 top4(35) -> 5 1097.04/291.53 active5(31) -> 36 1097.04/291.53 cons5(36, 30) -> 35 1097.04/291.53 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (6) 1097.04/291.53 BOUNDS(1, n^1) 1097.04/291.53 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1097.04/291.53 Transformed a relative TRS into a decreasing-loop problem. 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (8) 1097.04/291.53 Obligation: 1097.04/291.53 Analyzing the following TRS for decreasing loops: 1097.04/291.53 1097.04/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1097.04/291.53 1097.04/291.53 1097.04/291.53 The TRS R consists of the following rules: 1097.04/291.53 1097.04/291.53 active(zeros) -> mark(cons(0, zeros)) 1097.04/291.53 active(tail(cons(X, XS))) -> mark(XS) 1097.04/291.53 active(cons(X1, X2)) -> cons(active(X1), X2) 1097.04/291.53 active(tail(X)) -> tail(active(X)) 1097.04/291.53 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1097.04/291.53 tail(mark(X)) -> mark(tail(X)) 1097.04/291.53 proper(zeros) -> ok(zeros) 1097.04/291.53 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1097.04/291.53 proper(0) -> ok(0) 1097.04/291.53 proper(tail(X)) -> tail(proper(X)) 1097.04/291.53 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1097.04/291.53 tail(ok(X)) -> ok(tail(X)) 1097.04/291.53 top(mark(X)) -> top(proper(X)) 1097.04/291.53 top(ok(X)) -> top(active(X)) 1097.04/291.53 1097.04/291.53 S is empty. 1097.04/291.53 Rewrite Strategy: FULL 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (9) DecreasingLoopProof (LOWER BOUND(ID)) 1097.04/291.53 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1097.04/291.53 1097.04/291.53 The rewrite sequence 1097.04/291.53 1097.04/291.53 tail(ok(X)) ->^+ ok(tail(X)) 1097.04/291.53 1097.04/291.53 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1097.04/291.53 1097.04/291.53 The pumping substitution is [X / ok(X)]. 1097.04/291.53 1097.04/291.53 The result substitution is [ ]. 1097.04/291.53 1097.04/291.53 1097.04/291.53 1097.04/291.53 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (10) 1097.04/291.53 Complex Obligation (BEST) 1097.04/291.53 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (11) 1097.04/291.53 Obligation: 1097.04/291.53 Proved the lower bound n^1 for the following obligation: 1097.04/291.53 1097.04/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1097.04/291.53 1097.04/291.53 1097.04/291.53 The TRS R consists of the following rules: 1097.04/291.53 1097.04/291.53 active(zeros) -> mark(cons(0, zeros)) 1097.04/291.53 active(tail(cons(X, XS))) -> mark(XS) 1097.04/291.53 active(cons(X1, X2)) -> cons(active(X1), X2) 1097.04/291.53 active(tail(X)) -> tail(active(X)) 1097.04/291.53 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1097.04/291.53 tail(mark(X)) -> mark(tail(X)) 1097.04/291.53 proper(zeros) -> ok(zeros) 1097.04/291.53 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1097.04/291.53 proper(0) -> ok(0) 1097.04/291.53 proper(tail(X)) -> tail(proper(X)) 1097.04/291.53 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1097.04/291.53 tail(ok(X)) -> ok(tail(X)) 1097.04/291.53 top(mark(X)) -> top(proper(X)) 1097.04/291.53 top(ok(X)) -> top(active(X)) 1097.04/291.53 1097.04/291.53 S is empty. 1097.04/291.53 Rewrite Strategy: FULL 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (12) LowerBoundPropagationProof (FINISHED) 1097.04/291.53 Propagated lower bound. 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (13) 1097.04/291.53 BOUNDS(n^1, INF) 1097.04/291.53 1097.04/291.53 ---------------------------------------- 1097.04/291.53 1097.04/291.53 (14) 1097.04/291.53 Obligation: 1097.04/291.53 Analyzing the following TRS for decreasing loops: 1097.04/291.53 1097.04/291.53 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1097.04/291.53 1097.04/291.53 1097.04/291.53 The TRS R consists of the following rules: 1097.04/291.53 1097.04/291.53 active(zeros) -> mark(cons(0, zeros)) 1097.04/291.53 active(tail(cons(X, XS))) -> mark(XS) 1097.04/291.53 active(cons(X1, X2)) -> cons(active(X1), X2) 1097.04/291.53 active(tail(X)) -> tail(active(X)) 1097.04/291.53 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1097.04/291.53 tail(mark(X)) -> mark(tail(X)) 1097.04/291.53 proper(zeros) -> ok(zeros) 1097.04/291.53 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1097.04/291.53 proper(0) -> ok(0) 1097.04/291.53 proper(tail(X)) -> tail(proper(X)) 1097.04/291.53 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1097.04/291.53 tail(ok(X)) -> ok(tail(X)) 1097.04/291.53 top(mark(X)) -> top(proper(X)) 1097.04/291.53 top(ok(X)) -> top(active(X)) 1097.04/291.53 1097.04/291.53 S is empty. 1097.04/291.53 Rewrite Strategy: FULL 1097.14/291.60 EOF