21.78/7.33 WORST_CASE(Omega(n^1), O(n^1)) 21.78/7.34 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 21.78/7.34 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 21.78/7.34 21.78/7.34 21.78/7.34 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.78/7.34 21.78/7.34 (0) CpxTRS 21.78/7.34 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 21.78/7.34 (2) CpxTRS 21.78/7.34 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 21.78/7.34 (4) CpxTRS 21.78/7.34 (5) CpxTrsMatchBoundsTAProof [FINISHED, 26 ms] 21.78/7.34 (6) BOUNDS(1, n^1) 21.78/7.34 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 21.78/7.34 (8) TRS for Loop Detection 21.78/7.34 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 21.78/7.34 (10) BEST 21.78/7.34 (11) proven lower bound 21.78/7.34 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 21.78/7.34 (13) BOUNDS(n^1, INF) 21.78/7.34 (14) TRS for Loop Detection 21.78/7.34 21.78/7.34 21.78/7.34 ---------------------------------------- 21.78/7.34 21.78/7.34 (0) 21.78/7.34 Obligation: 21.78/7.34 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.78/7.34 21.78/7.34 21.78/7.34 The TRS R consists of the following rules: 21.78/7.34 21.78/7.34 active(f(0)) -> mark(cons(0, f(s(0)))) 21.78/7.34 active(f(s(0))) -> mark(f(p(s(0)))) 21.78/7.34 active(p(s(X))) -> mark(X) 21.78/7.34 active(f(X)) -> f(active(X)) 21.78/7.34 active(cons(X1, X2)) -> cons(active(X1), X2) 21.78/7.34 active(s(X)) -> s(active(X)) 21.78/7.34 active(p(X)) -> p(active(X)) 21.78/7.34 f(mark(X)) -> mark(f(X)) 21.78/7.34 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.78/7.34 s(mark(X)) -> mark(s(X)) 21.78/7.34 p(mark(X)) -> mark(p(X)) 21.78/7.34 proper(f(X)) -> f(proper(X)) 21.78/7.34 proper(0) -> ok(0) 21.78/7.34 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.78/7.34 proper(s(X)) -> s(proper(X)) 21.78/7.34 proper(p(X)) -> p(proper(X)) 21.78/7.34 f(ok(X)) -> ok(f(X)) 21.78/7.34 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.78/7.34 s(ok(X)) -> ok(s(X)) 21.78/7.34 p(ok(X)) -> ok(p(X)) 21.78/7.34 top(mark(X)) -> top(proper(X)) 21.78/7.34 top(ok(X)) -> top(active(X)) 21.78/7.34 21.78/7.34 S is empty. 21.78/7.34 Rewrite Strategy: FULL 21.78/7.34 ---------------------------------------- 21.78/7.34 21.78/7.34 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 21.78/7.34 The following defined symbols can occur below the 0th argument of top: proper, active 21.78/7.34 The following defined symbols can occur below the 0th argument of proper: proper, active 21.78/7.34 The following defined symbols can occur below the 0th argument of active: proper, active 21.78/7.34 21.78/7.34 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 21.78/7.34 active(f(0)) -> mark(cons(0, f(s(0)))) 21.78/7.34 active(f(s(0))) -> mark(f(p(s(0)))) 21.78/7.34 active(p(s(X))) -> mark(X) 21.78/7.34 active(f(X)) -> f(active(X)) 21.78/7.34 active(cons(X1, X2)) -> cons(active(X1), X2) 21.78/7.34 active(s(X)) -> s(active(X)) 21.78/7.34 active(p(X)) -> p(active(X)) 21.78/7.34 proper(f(X)) -> f(proper(X)) 21.78/7.34 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.78/7.34 proper(s(X)) -> s(proper(X)) 21.78/7.34 proper(p(X)) -> p(proper(X)) 21.78/7.34 21.78/7.34 ---------------------------------------- 21.78/7.34 21.78/7.34 (2) 21.78/7.34 Obligation: 21.78/7.34 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 21.78/7.34 21.78/7.34 21.78/7.34 The TRS R consists of the following rules: 21.78/7.34 21.78/7.34 f(mark(X)) -> mark(f(X)) 21.78/7.34 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.78/7.34 s(mark(X)) -> mark(s(X)) 21.78/7.34 p(mark(X)) -> mark(p(X)) 21.78/7.34 proper(0) -> ok(0) 21.78/7.34 f(ok(X)) -> ok(f(X)) 21.78/7.34 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.78/7.34 s(ok(X)) -> ok(s(X)) 21.78/7.34 p(ok(X)) -> ok(p(X)) 21.78/7.34 top(mark(X)) -> top(proper(X)) 21.78/7.34 top(ok(X)) -> top(active(X)) 21.78/7.35 21.78/7.35 S is empty. 21.78/7.35 Rewrite Strategy: FULL 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 21.78/7.35 transformed relative TRS to TRS 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (4) 21.78/7.35 Obligation: 21.78/7.35 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 21.78/7.35 21.78/7.35 21.78/7.35 The TRS R consists of the following rules: 21.78/7.35 21.78/7.35 f(mark(X)) -> mark(f(X)) 21.78/7.35 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.78/7.35 s(mark(X)) -> mark(s(X)) 21.78/7.35 p(mark(X)) -> mark(p(X)) 21.78/7.35 proper(0) -> ok(0) 21.78/7.35 f(ok(X)) -> ok(f(X)) 21.78/7.35 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.78/7.35 s(ok(X)) -> ok(s(X)) 21.78/7.35 p(ok(X)) -> ok(p(X)) 21.78/7.35 top(mark(X)) -> top(proper(X)) 21.78/7.35 top(ok(X)) -> top(active(X)) 21.78/7.35 21.78/7.35 S is empty. 21.78/7.35 Rewrite Strategy: FULL 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (5) CpxTrsMatchBoundsTAProof (FINISHED) 21.78/7.35 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. 21.78/7.35 21.78/7.35 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 21.78/7.35 final states : [1, 2, 3, 4, 5, 6] 21.78/7.35 transitions: 21.78/7.35 mark0(0) -> 0 21.78/7.35 00() -> 0 21.78/7.35 ok0(0) -> 0 21.78/7.35 active0(0) -> 0 21.78/7.35 f0(0) -> 1 21.78/7.35 cons0(0, 0) -> 2 21.78/7.35 s0(0) -> 3 21.78/7.35 p0(0) -> 4 21.78/7.35 proper0(0) -> 5 21.78/7.35 top0(0) -> 6 21.78/7.35 f1(0) -> 7 21.78/7.35 mark1(7) -> 1 21.78/7.35 cons1(0, 0) -> 8 21.78/7.35 mark1(8) -> 2 21.78/7.35 s1(0) -> 9 21.78/7.35 mark1(9) -> 3 21.78/7.35 p1(0) -> 10 21.78/7.35 mark1(10) -> 4 21.78/7.35 01() -> 11 21.78/7.35 ok1(11) -> 5 21.78/7.35 f1(0) -> 12 21.78/7.35 ok1(12) -> 1 21.78/7.35 cons1(0, 0) -> 13 21.78/7.35 ok1(13) -> 2 21.78/7.35 s1(0) -> 14 21.78/7.35 ok1(14) -> 3 21.78/7.35 p1(0) -> 15 21.78/7.35 ok1(15) -> 4 21.78/7.35 proper1(0) -> 16 21.78/7.35 top1(16) -> 6 21.78/7.35 active1(0) -> 17 21.78/7.35 top1(17) -> 6 21.78/7.35 mark1(7) -> 7 21.78/7.35 mark1(7) -> 12 21.78/7.35 mark1(8) -> 8 21.78/7.35 mark1(8) -> 13 21.78/7.35 mark1(9) -> 9 21.78/7.35 mark1(9) -> 14 21.78/7.35 mark1(10) -> 10 21.78/7.35 mark1(10) -> 15 21.78/7.35 ok1(11) -> 16 21.78/7.35 ok1(12) -> 7 21.78/7.35 ok1(12) -> 12 21.78/7.35 ok1(13) -> 8 21.78/7.35 ok1(13) -> 13 21.78/7.35 ok1(14) -> 9 21.78/7.35 ok1(14) -> 14 21.78/7.35 ok1(15) -> 10 21.78/7.35 ok1(15) -> 15 21.78/7.35 active2(11) -> 18 21.78/7.35 top2(18) -> 6 21.78/7.35 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (6) 21.78/7.35 BOUNDS(1, n^1) 21.78/7.35 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 21.78/7.35 Transformed a relative TRS into a decreasing-loop problem. 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (8) 21.78/7.35 Obligation: 21.78/7.35 Analyzing the following TRS for decreasing loops: 21.78/7.35 21.78/7.35 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.78/7.35 21.78/7.35 21.78/7.35 The TRS R consists of the following rules: 21.78/7.35 21.78/7.35 active(f(0)) -> mark(cons(0, f(s(0)))) 21.78/7.35 active(f(s(0))) -> mark(f(p(s(0)))) 21.78/7.35 active(p(s(X))) -> mark(X) 21.78/7.35 active(f(X)) -> f(active(X)) 21.78/7.35 active(cons(X1, X2)) -> cons(active(X1), X2) 21.78/7.35 active(s(X)) -> s(active(X)) 21.78/7.35 active(p(X)) -> p(active(X)) 21.78/7.35 f(mark(X)) -> mark(f(X)) 21.78/7.35 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.78/7.35 s(mark(X)) -> mark(s(X)) 21.78/7.35 p(mark(X)) -> mark(p(X)) 21.78/7.35 proper(f(X)) -> f(proper(X)) 21.78/7.35 proper(0) -> ok(0) 21.78/7.35 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.78/7.35 proper(s(X)) -> s(proper(X)) 21.78/7.35 proper(p(X)) -> p(proper(X)) 21.78/7.35 f(ok(X)) -> ok(f(X)) 21.78/7.35 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.78/7.35 s(ok(X)) -> ok(s(X)) 21.78/7.35 p(ok(X)) -> ok(p(X)) 21.78/7.35 top(mark(X)) -> top(proper(X)) 21.78/7.35 top(ok(X)) -> top(active(X)) 21.78/7.35 21.78/7.35 S is empty. 21.78/7.35 Rewrite Strategy: FULL 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (9) DecreasingLoopProof (LOWER BOUND(ID)) 21.78/7.35 The following loop(s) give(s) rise to the lower bound Omega(n^1): 21.78/7.35 21.78/7.35 The rewrite sequence 21.78/7.35 21.78/7.35 p(mark(X)) ->^+ mark(p(X)) 21.78/7.35 21.78/7.35 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 21.78/7.35 21.78/7.35 The pumping substitution is [X / mark(X)]. 21.78/7.35 21.78/7.35 The result substitution is [ ]. 21.78/7.35 21.78/7.35 21.78/7.35 21.78/7.35 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (10) 21.78/7.35 Complex Obligation (BEST) 21.78/7.35 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (11) 21.78/7.35 Obligation: 21.78/7.35 Proved the lower bound n^1 for the following obligation: 21.78/7.35 21.78/7.35 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.78/7.35 21.78/7.35 21.78/7.35 The TRS R consists of the following rules: 21.78/7.35 21.78/7.35 active(f(0)) -> mark(cons(0, f(s(0)))) 21.78/7.35 active(f(s(0))) -> mark(f(p(s(0)))) 21.78/7.35 active(p(s(X))) -> mark(X) 21.78/7.35 active(f(X)) -> f(active(X)) 21.78/7.35 active(cons(X1, X2)) -> cons(active(X1), X2) 21.78/7.35 active(s(X)) -> s(active(X)) 21.78/7.35 active(p(X)) -> p(active(X)) 21.78/7.35 f(mark(X)) -> mark(f(X)) 21.78/7.35 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.78/7.35 s(mark(X)) -> mark(s(X)) 21.78/7.35 p(mark(X)) -> mark(p(X)) 21.78/7.35 proper(f(X)) -> f(proper(X)) 21.78/7.35 proper(0) -> ok(0) 21.78/7.35 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.78/7.35 proper(s(X)) -> s(proper(X)) 21.78/7.35 proper(p(X)) -> p(proper(X)) 21.78/7.35 f(ok(X)) -> ok(f(X)) 21.78/7.35 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.78/7.35 s(ok(X)) -> ok(s(X)) 21.78/7.35 p(ok(X)) -> ok(p(X)) 21.78/7.35 top(mark(X)) -> top(proper(X)) 21.78/7.35 top(ok(X)) -> top(active(X)) 21.78/7.35 21.78/7.35 S is empty. 21.78/7.35 Rewrite Strategy: FULL 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (12) LowerBoundPropagationProof (FINISHED) 21.78/7.35 Propagated lower bound. 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (13) 21.78/7.35 BOUNDS(n^1, INF) 21.78/7.35 21.78/7.35 ---------------------------------------- 21.78/7.35 21.78/7.35 (14) 21.78/7.35 Obligation: 21.78/7.35 Analyzing the following TRS for decreasing loops: 21.78/7.35 21.78/7.35 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.78/7.35 21.78/7.35 21.78/7.35 The TRS R consists of the following rules: 21.78/7.35 21.78/7.35 active(f(0)) -> mark(cons(0, f(s(0)))) 21.78/7.35 active(f(s(0))) -> mark(f(p(s(0)))) 21.78/7.35 active(p(s(X))) -> mark(X) 21.78/7.35 active(f(X)) -> f(active(X)) 21.78/7.35 active(cons(X1, X2)) -> cons(active(X1), X2) 21.78/7.35 active(s(X)) -> s(active(X)) 21.78/7.35 active(p(X)) -> p(active(X)) 21.78/7.35 f(mark(X)) -> mark(f(X)) 21.78/7.35 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.78/7.35 s(mark(X)) -> mark(s(X)) 21.78/7.35 p(mark(X)) -> mark(p(X)) 21.78/7.35 proper(f(X)) -> f(proper(X)) 21.78/7.35 proper(0) -> ok(0) 21.78/7.35 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.78/7.35 proper(s(X)) -> s(proper(X)) 21.78/7.35 proper(p(X)) -> p(proper(X)) 21.78/7.35 f(ok(X)) -> ok(f(X)) 21.78/7.35 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.78/7.35 s(ok(X)) -> ok(s(X)) 21.78/7.35 p(ok(X)) -> ok(p(X)) 21.78/7.35 top(mark(X)) -> top(proper(X)) 21.78/7.35 top(ok(X)) -> top(active(X)) 21.78/7.35 21.78/7.35 S is empty. 21.78/7.35 Rewrite Strategy: FULL 22.00/7.40 EOF