25.59/9.61 WORST_CASE(Omega(n^1), O(n^1)) 25.59/9.62 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 25.59/9.62 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.59/9.62 25.59/9.62 25.59/9.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.59/9.62 25.59/9.62 (0) CpxTRS 25.59/9.62 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 25.59/9.62 (2) CpxTRS 25.59/9.62 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 25.59/9.62 (4) CpxTRS 25.59/9.62 (5) CpxTrsMatchBoundsTAProof [FINISHED, 46 ms] 25.59/9.62 (6) BOUNDS(1, n^1) 25.59/9.62 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 25.59/9.62 (8) TRS for Loop Detection 25.59/9.62 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 25.59/9.62 (10) BEST 25.59/9.62 (11) proven lower bound 25.59/9.62 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 25.59/9.62 (13) BOUNDS(n^1, INF) 25.59/9.62 (14) TRS for Loop Detection 25.59/9.62 25.59/9.62 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (0) 25.59/9.62 Obligation: 25.59/9.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.59/9.62 25.59/9.62 25.59/9.62 The TRS R consists of the following rules: 25.59/9.62 25.59/9.62 active(from(X)) -> mark(cons(X, from(s(X)))) 25.59/9.62 active(after(0, XS)) -> mark(XS) 25.59/9.62 active(after(s(N), cons(X, XS))) -> mark(after(N, XS)) 25.59/9.62 active(from(X)) -> from(active(X)) 25.59/9.62 active(cons(X1, X2)) -> cons(active(X1), X2) 25.59/9.62 active(s(X)) -> s(active(X)) 25.59/9.62 active(after(X1, X2)) -> after(active(X1), X2) 25.59/9.62 active(after(X1, X2)) -> after(X1, active(X2)) 25.59/9.62 from(mark(X)) -> mark(from(X)) 25.59/9.62 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.59/9.62 s(mark(X)) -> mark(s(X)) 25.59/9.62 after(mark(X1), X2) -> mark(after(X1, X2)) 25.59/9.62 after(X1, mark(X2)) -> mark(after(X1, X2)) 25.59/9.62 proper(from(X)) -> from(proper(X)) 25.59/9.62 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.59/9.62 proper(s(X)) -> s(proper(X)) 25.59/9.62 proper(after(X1, X2)) -> after(proper(X1), proper(X2)) 25.59/9.62 proper(0) -> ok(0) 25.59/9.62 from(ok(X)) -> ok(from(X)) 25.59/9.62 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.59/9.62 s(ok(X)) -> ok(s(X)) 25.59/9.62 after(ok(X1), ok(X2)) -> ok(after(X1, X2)) 25.59/9.62 top(mark(X)) -> top(proper(X)) 25.59/9.62 top(ok(X)) -> top(active(X)) 25.59/9.62 25.59/9.62 S is empty. 25.59/9.62 Rewrite Strategy: FULL 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 25.59/9.62 The following defined symbols can occur below the 0th argument of top: proper, active 25.59/9.62 The following defined symbols can occur below the 0th argument of proper: proper, active 25.59/9.62 The following defined symbols can occur below the 0th argument of active: proper, active 25.59/9.62 25.59/9.62 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 25.59/9.62 active(from(X)) -> mark(cons(X, from(s(X)))) 25.59/9.62 active(after(0, XS)) -> mark(XS) 25.59/9.62 active(after(s(N), cons(X, XS))) -> mark(after(N, XS)) 25.59/9.62 active(from(X)) -> from(active(X)) 25.59/9.62 active(cons(X1, X2)) -> cons(active(X1), X2) 25.59/9.62 active(s(X)) -> s(active(X)) 25.59/9.62 active(after(X1, X2)) -> after(active(X1), X2) 25.59/9.62 active(after(X1, X2)) -> after(X1, active(X2)) 25.59/9.62 proper(from(X)) -> from(proper(X)) 25.59/9.62 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.59/9.62 proper(s(X)) -> s(proper(X)) 25.59/9.62 proper(after(X1, X2)) -> after(proper(X1), proper(X2)) 25.59/9.62 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (2) 25.59/9.62 Obligation: 25.59/9.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 25.59/9.62 25.59/9.62 25.59/9.62 The TRS R consists of the following rules: 25.59/9.62 25.59/9.62 from(mark(X)) -> mark(from(X)) 25.59/9.62 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.59/9.62 s(mark(X)) -> mark(s(X)) 25.59/9.62 after(mark(X1), X2) -> mark(after(X1, X2)) 25.59/9.62 after(X1, mark(X2)) -> mark(after(X1, X2)) 25.59/9.62 proper(0) -> ok(0) 25.59/9.62 from(ok(X)) -> ok(from(X)) 25.59/9.62 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.59/9.62 s(ok(X)) -> ok(s(X)) 25.59/9.62 after(ok(X1), ok(X2)) -> ok(after(X1, X2)) 25.59/9.62 top(mark(X)) -> top(proper(X)) 25.59/9.62 top(ok(X)) -> top(active(X)) 25.59/9.62 25.59/9.62 S is empty. 25.59/9.62 Rewrite Strategy: FULL 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 25.59/9.62 transformed relative TRS to TRS 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (4) 25.59/9.62 Obligation: 25.59/9.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 25.59/9.62 25.59/9.62 25.59/9.62 The TRS R consists of the following rules: 25.59/9.62 25.59/9.62 from(mark(X)) -> mark(from(X)) 25.59/9.62 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.59/9.62 s(mark(X)) -> mark(s(X)) 25.59/9.62 after(mark(X1), X2) -> mark(after(X1, X2)) 25.59/9.62 after(X1, mark(X2)) -> mark(after(X1, X2)) 25.59/9.62 proper(0) -> ok(0) 25.59/9.62 from(ok(X)) -> ok(from(X)) 25.59/9.62 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.59/9.62 s(ok(X)) -> ok(s(X)) 25.59/9.62 after(ok(X1), ok(X2)) -> ok(after(X1, X2)) 25.59/9.62 top(mark(X)) -> top(proper(X)) 25.59/9.62 top(ok(X)) -> top(active(X)) 25.59/9.62 25.59/9.62 S is empty. 25.59/9.62 Rewrite Strategy: FULL 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (5) CpxTrsMatchBoundsTAProof (FINISHED) 25.59/9.62 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. 25.59/9.62 25.59/9.62 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 25.59/9.62 final states : [1, 2, 3, 4, 5, 6] 25.59/9.62 transitions: 25.59/9.62 mark0(0) -> 0 25.59/9.62 00() -> 0 25.59/9.62 ok0(0) -> 0 25.59/9.62 active0(0) -> 0 25.59/9.62 from0(0) -> 1 25.59/9.62 cons0(0, 0) -> 2 25.59/9.62 s0(0) -> 3 25.59/9.62 after0(0, 0) -> 4 25.59/9.62 proper0(0) -> 5 25.59/9.62 top0(0) -> 6 25.59/9.62 from1(0) -> 7 25.59/9.62 mark1(7) -> 1 25.59/9.62 cons1(0, 0) -> 8 25.59/9.62 mark1(8) -> 2 25.59/9.62 s1(0) -> 9 25.59/9.62 mark1(9) -> 3 25.59/9.62 after1(0, 0) -> 10 25.59/9.62 mark1(10) -> 4 25.59/9.62 01() -> 11 25.59/9.62 ok1(11) -> 5 25.59/9.62 from1(0) -> 12 25.59/9.62 ok1(12) -> 1 25.59/9.62 cons1(0, 0) -> 13 25.59/9.62 ok1(13) -> 2 25.59/9.62 s1(0) -> 14 25.59/9.62 ok1(14) -> 3 25.59/9.62 after1(0, 0) -> 15 25.59/9.62 ok1(15) -> 4 25.59/9.62 proper1(0) -> 16 25.59/9.62 top1(16) -> 6 25.59/9.62 active1(0) -> 17 25.59/9.62 top1(17) -> 6 25.59/9.62 mark1(7) -> 7 25.59/9.62 mark1(7) -> 12 25.59/9.62 mark1(8) -> 8 25.59/9.62 mark1(8) -> 13 25.59/9.62 mark1(9) -> 9 25.59/9.62 mark1(9) -> 14 25.59/9.62 mark1(10) -> 10 25.59/9.62 mark1(10) -> 15 25.59/9.62 ok1(11) -> 16 25.59/9.62 ok1(12) -> 7 25.59/9.62 ok1(12) -> 12 25.59/9.62 ok1(13) -> 8 25.59/9.62 ok1(13) -> 13 25.59/9.62 ok1(14) -> 9 25.59/9.62 ok1(14) -> 14 25.59/9.62 ok1(15) -> 10 25.59/9.62 ok1(15) -> 15 25.59/9.62 active2(11) -> 18 25.59/9.62 top2(18) -> 6 25.59/9.62 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (6) 25.59/9.62 BOUNDS(1, n^1) 25.59/9.62 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 25.59/9.62 Transformed a relative TRS into a decreasing-loop problem. 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (8) 25.59/9.62 Obligation: 25.59/9.62 Analyzing the following TRS for decreasing loops: 25.59/9.62 25.59/9.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.59/9.62 25.59/9.62 25.59/9.62 The TRS R consists of the following rules: 25.59/9.62 25.59/9.62 active(from(X)) -> mark(cons(X, from(s(X)))) 25.59/9.62 active(after(0, XS)) -> mark(XS) 25.59/9.62 active(after(s(N), cons(X, XS))) -> mark(after(N, XS)) 25.59/9.62 active(from(X)) -> from(active(X)) 25.59/9.62 active(cons(X1, X2)) -> cons(active(X1), X2) 25.59/9.62 active(s(X)) -> s(active(X)) 25.59/9.62 active(after(X1, X2)) -> after(active(X1), X2) 25.59/9.62 active(after(X1, X2)) -> after(X1, active(X2)) 25.59/9.62 from(mark(X)) -> mark(from(X)) 25.59/9.62 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.59/9.62 s(mark(X)) -> mark(s(X)) 25.59/9.62 after(mark(X1), X2) -> mark(after(X1, X2)) 25.59/9.62 after(X1, mark(X2)) -> mark(after(X1, X2)) 25.59/9.62 proper(from(X)) -> from(proper(X)) 25.59/9.62 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.59/9.62 proper(s(X)) -> s(proper(X)) 25.59/9.62 proper(after(X1, X2)) -> after(proper(X1), proper(X2)) 25.59/9.62 proper(0) -> ok(0) 25.59/9.62 from(ok(X)) -> ok(from(X)) 25.59/9.62 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.59/9.62 s(ok(X)) -> ok(s(X)) 25.59/9.62 after(ok(X1), ok(X2)) -> ok(after(X1, X2)) 25.59/9.62 top(mark(X)) -> top(proper(X)) 25.59/9.62 top(ok(X)) -> top(active(X)) 25.59/9.62 25.59/9.62 S is empty. 25.59/9.62 Rewrite Strategy: FULL 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (9) DecreasingLoopProof (LOWER BOUND(ID)) 25.59/9.62 The following loop(s) give(s) rise to the lower bound Omega(n^1): 25.59/9.62 25.59/9.62 The rewrite sequence 25.59/9.62 25.59/9.62 after(ok(X1), ok(X2)) ->^+ ok(after(X1, X2)) 25.59/9.62 25.59/9.62 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 25.59/9.62 25.59/9.62 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 25.59/9.62 25.59/9.62 The result substitution is [ ]. 25.59/9.62 25.59/9.62 25.59/9.62 25.59/9.62 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (10) 25.59/9.62 Complex Obligation (BEST) 25.59/9.62 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (11) 25.59/9.62 Obligation: 25.59/9.62 Proved the lower bound n^1 for the following obligation: 25.59/9.62 25.59/9.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.59/9.62 25.59/9.62 25.59/9.62 The TRS R consists of the following rules: 25.59/9.62 25.59/9.62 active(from(X)) -> mark(cons(X, from(s(X)))) 25.59/9.62 active(after(0, XS)) -> mark(XS) 25.59/9.62 active(after(s(N), cons(X, XS))) -> mark(after(N, XS)) 25.59/9.62 active(from(X)) -> from(active(X)) 25.59/9.62 active(cons(X1, X2)) -> cons(active(X1), X2) 25.59/9.62 active(s(X)) -> s(active(X)) 25.59/9.62 active(after(X1, X2)) -> after(active(X1), X2) 25.59/9.62 active(after(X1, X2)) -> after(X1, active(X2)) 25.59/9.62 from(mark(X)) -> mark(from(X)) 25.59/9.62 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.59/9.62 s(mark(X)) -> mark(s(X)) 25.59/9.62 after(mark(X1), X2) -> mark(after(X1, X2)) 25.59/9.62 after(X1, mark(X2)) -> mark(after(X1, X2)) 25.59/9.62 proper(from(X)) -> from(proper(X)) 25.59/9.62 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.59/9.62 proper(s(X)) -> s(proper(X)) 25.59/9.62 proper(after(X1, X2)) -> after(proper(X1), proper(X2)) 25.59/9.62 proper(0) -> ok(0) 25.59/9.62 from(ok(X)) -> ok(from(X)) 25.59/9.62 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.59/9.62 s(ok(X)) -> ok(s(X)) 25.59/9.62 after(ok(X1), ok(X2)) -> ok(after(X1, X2)) 25.59/9.62 top(mark(X)) -> top(proper(X)) 25.59/9.62 top(ok(X)) -> top(active(X)) 25.59/9.62 25.59/9.62 S is empty. 25.59/9.62 Rewrite Strategy: FULL 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (12) LowerBoundPropagationProof (FINISHED) 25.59/9.62 Propagated lower bound. 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (13) 25.59/9.62 BOUNDS(n^1, INF) 25.59/9.62 25.59/9.62 ---------------------------------------- 25.59/9.62 25.59/9.62 (14) 25.59/9.62 Obligation: 25.59/9.62 Analyzing the following TRS for decreasing loops: 25.59/9.62 25.59/9.62 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.59/9.62 25.59/9.62 25.59/9.62 The TRS R consists of the following rules: 25.59/9.62 25.59/9.62 active(from(X)) -> mark(cons(X, from(s(X)))) 25.59/9.62 active(after(0, XS)) -> mark(XS) 25.59/9.62 active(after(s(N), cons(X, XS))) -> mark(after(N, XS)) 25.59/9.62 active(from(X)) -> from(active(X)) 25.59/9.62 active(cons(X1, X2)) -> cons(active(X1), X2) 25.59/9.62 active(s(X)) -> s(active(X)) 25.59/9.62 active(after(X1, X2)) -> after(active(X1), X2) 25.59/9.62 active(after(X1, X2)) -> after(X1, active(X2)) 25.59/9.62 from(mark(X)) -> mark(from(X)) 25.59/9.62 cons(mark(X1), X2) -> mark(cons(X1, X2)) 25.59/9.62 s(mark(X)) -> mark(s(X)) 25.59/9.62 after(mark(X1), X2) -> mark(after(X1, X2)) 25.59/9.62 after(X1, mark(X2)) -> mark(after(X1, X2)) 25.59/9.62 proper(from(X)) -> from(proper(X)) 25.59/9.62 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 25.59/9.62 proper(s(X)) -> s(proper(X)) 25.59/9.62 proper(after(X1, X2)) -> after(proper(X1), proper(X2)) 25.59/9.62 proper(0) -> ok(0) 25.59/9.62 from(ok(X)) -> ok(from(X)) 25.59/9.62 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 25.59/9.62 s(ok(X)) -> ok(s(X)) 25.59/9.62 after(ok(X1), ok(X2)) -> ok(after(X1, X2)) 25.59/9.62 top(mark(X)) -> top(proper(X)) 25.59/9.62 top(ok(X)) -> top(active(X)) 25.59/9.62 25.59/9.62 S is empty. 25.59/9.62 Rewrite Strategy: FULL 25.80/9.66 EOF