24.76/8.80 WORST_CASE(Omega(n^1), O(n^1)) 24.76/8.81 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.76/8.81 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.76/8.81 24.76/8.81 24.76/8.81 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.76/8.81 24.76/8.81 (0) CpxTRS 24.76/8.81 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 24.76/8.81 (2) CpxTRS 24.76/8.81 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.76/8.81 (4) CpxTRS 24.76/8.81 (5) CpxTrsMatchBoundsTAProof [FINISHED, 52 ms] 24.76/8.81 (6) BOUNDS(1, n^1) 24.76/8.81 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 24.76/8.81 (8) TRS for Loop Detection 24.76/8.81 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 24.76/8.81 (10) BEST 24.76/8.81 (11) proven lower bound 24.76/8.81 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 24.76/8.81 (13) BOUNDS(n^1, INF) 24.76/8.81 (14) TRS for Loop Detection 24.76/8.81 24.76/8.81 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (0) 24.76/8.81 Obligation: 24.76/8.81 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.76/8.81 24.76/8.81 24.76/8.81 The TRS R consists of the following rules: 24.76/8.81 24.76/8.81 active(from(X)) -> mark(cons(X, from(s(X)))) 24.76/8.81 active(sel(0, cons(X, Y))) -> mark(X) 24.76/8.81 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 24.76/8.81 active(from(X)) -> from(active(X)) 24.76/8.81 active(cons(X1, X2)) -> cons(active(X1), X2) 24.76/8.81 active(s(X)) -> s(active(X)) 24.76/8.81 active(sel(X1, X2)) -> sel(active(X1), X2) 24.76/8.81 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.76/8.81 from(mark(X)) -> mark(from(X)) 24.76/8.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.76/8.81 s(mark(X)) -> mark(s(X)) 24.76/8.81 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.76/8.81 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.76/8.81 proper(from(X)) -> from(proper(X)) 24.76/8.81 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.76/8.81 proper(s(X)) -> s(proper(X)) 24.76/8.81 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.76/8.81 proper(0) -> ok(0) 24.76/8.81 from(ok(X)) -> ok(from(X)) 24.76/8.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.76/8.81 s(ok(X)) -> ok(s(X)) 24.76/8.81 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.76/8.81 top(mark(X)) -> top(proper(X)) 24.76/8.81 top(ok(X)) -> top(active(X)) 24.76/8.81 24.76/8.81 S is empty. 24.76/8.81 Rewrite Strategy: FULL 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.76/8.81 The following defined symbols can occur below the 0th argument of top: proper, active 24.76/8.81 The following defined symbols can occur below the 0th argument of proper: proper, active 24.76/8.81 The following defined symbols can occur below the 0th argument of active: proper, active 24.76/8.81 24.76/8.81 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.76/8.81 active(from(X)) -> mark(cons(X, from(s(X)))) 24.76/8.81 active(sel(0, cons(X, Y))) -> mark(X) 24.76/8.81 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 24.76/8.81 active(from(X)) -> from(active(X)) 24.76/8.81 active(cons(X1, X2)) -> cons(active(X1), X2) 24.76/8.81 active(s(X)) -> s(active(X)) 24.76/8.81 active(sel(X1, X2)) -> sel(active(X1), X2) 24.76/8.81 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.76/8.81 proper(from(X)) -> from(proper(X)) 24.76/8.81 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.76/8.81 proper(s(X)) -> s(proper(X)) 24.76/8.81 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.76/8.81 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (2) 24.76/8.81 Obligation: 24.76/8.81 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.76/8.81 24.76/8.81 24.76/8.81 The TRS R consists of the following rules: 24.76/8.81 24.76/8.81 from(mark(X)) -> mark(from(X)) 24.76/8.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.76/8.81 s(mark(X)) -> mark(s(X)) 24.76/8.81 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.76/8.81 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.76/8.81 proper(0) -> ok(0) 24.76/8.81 from(ok(X)) -> ok(from(X)) 24.76/8.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.76/8.81 s(ok(X)) -> ok(s(X)) 24.76/8.81 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.76/8.81 top(mark(X)) -> top(proper(X)) 24.76/8.81 top(ok(X)) -> top(active(X)) 24.76/8.81 24.76/8.81 S is empty. 24.76/8.81 Rewrite Strategy: FULL 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.76/8.81 transformed relative TRS to TRS 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (4) 24.76/8.81 Obligation: 24.76/8.81 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.76/8.81 24.76/8.81 24.76/8.81 The TRS R consists of the following rules: 24.76/8.81 24.76/8.81 from(mark(X)) -> mark(from(X)) 24.76/8.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.76/8.81 s(mark(X)) -> mark(s(X)) 24.76/8.81 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.76/8.81 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.76/8.81 proper(0) -> ok(0) 24.76/8.81 from(ok(X)) -> ok(from(X)) 24.76/8.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.76/8.81 s(ok(X)) -> ok(s(X)) 24.76/8.81 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.76/8.81 top(mark(X)) -> top(proper(X)) 24.76/8.81 top(ok(X)) -> top(active(X)) 24.76/8.81 24.76/8.81 S is empty. 24.76/8.81 Rewrite Strategy: FULL 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.76/8.81 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. 24.76/8.81 24.76/8.81 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.76/8.81 final states : [1, 2, 3, 4, 5, 6] 24.76/8.81 transitions: 24.76/8.81 mark0(0) -> 0 24.76/8.81 00() -> 0 24.76/8.81 ok0(0) -> 0 24.76/8.81 active0(0) -> 0 24.76/8.81 from0(0) -> 1 24.76/8.81 cons0(0, 0) -> 2 24.76/8.81 s0(0) -> 3 24.76/8.81 sel0(0, 0) -> 4 24.76/8.81 proper0(0) -> 5 24.76/8.81 top0(0) -> 6 24.76/8.81 from1(0) -> 7 24.76/8.81 mark1(7) -> 1 24.76/8.81 cons1(0, 0) -> 8 24.76/8.81 mark1(8) -> 2 24.76/8.81 s1(0) -> 9 24.76/8.81 mark1(9) -> 3 24.76/8.81 sel1(0, 0) -> 10 24.76/8.81 mark1(10) -> 4 24.76/8.81 01() -> 11 24.76/8.81 ok1(11) -> 5 24.76/8.81 from1(0) -> 12 24.76/8.81 ok1(12) -> 1 24.76/8.81 cons1(0, 0) -> 13 24.76/8.81 ok1(13) -> 2 24.76/8.81 s1(0) -> 14 24.76/8.81 ok1(14) -> 3 24.76/8.81 sel1(0, 0) -> 15 24.76/8.81 ok1(15) -> 4 24.76/8.81 proper1(0) -> 16 24.76/8.81 top1(16) -> 6 24.76/8.81 active1(0) -> 17 24.76/8.81 top1(17) -> 6 24.76/8.81 mark1(7) -> 7 24.76/8.81 mark1(7) -> 12 24.76/8.81 mark1(8) -> 8 24.76/8.81 mark1(8) -> 13 24.76/8.81 mark1(9) -> 9 24.76/8.81 mark1(9) -> 14 24.76/8.81 mark1(10) -> 10 24.76/8.81 mark1(10) -> 15 24.76/8.81 ok1(11) -> 16 24.76/8.81 ok1(12) -> 7 24.76/8.81 ok1(12) -> 12 24.76/8.81 ok1(13) -> 8 24.76/8.81 ok1(13) -> 13 24.76/8.81 ok1(14) -> 9 24.76/8.81 ok1(14) -> 14 24.76/8.81 ok1(15) -> 10 24.76/8.81 ok1(15) -> 15 24.76/8.81 active2(11) -> 18 24.76/8.81 top2(18) -> 6 24.76/8.81 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (6) 24.76/8.81 BOUNDS(1, n^1) 24.76/8.81 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 24.76/8.81 Transformed a relative TRS into a decreasing-loop problem. 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (8) 24.76/8.81 Obligation: 24.76/8.81 Analyzing the following TRS for decreasing loops: 24.76/8.81 24.76/8.81 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.76/8.81 24.76/8.81 24.76/8.81 The TRS R consists of the following rules: 24.76/8.81 24.76/8.81 active(from(X)) -> mark(cons(X, from(s(X)))) 24.76/8.81 active(sel(0, cons(X, Y))) -> mark(X) 24.76/8.81 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 24.76/8.81 active(from(X)) -> from(active(X)) 24.76/8.81 active(cons(X1, X2)) -> cons(active(X1), X2) 24.76/8.81 active(s(X)) -> s(active(X)) 24.76/8.81 active(sel(X1, X2)) -> sel(active(X1), X2) 24.76/8.81 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.76/8.81 from(mark(X)) -> mark(from(X)) 24.76/8.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.76/8.81 s(mark(X)) -> mark(s(X)) 24.76/8.81 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.76/8.81 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.76/8.81 proper(from(X)) -> from(proper(X)) 24.76/8.81 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.76/8.81 proper(s(X)) -> s(proper(X)) 24.76/8.81 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.76/8.81 proper(0) -> ok(0) 24.76/8.81 from(ok(X)) -> ok(from(X)) 24.76/8.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.76/8.81 s(ok(X)) -> ok(s(X)) 24.76/8.81 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.76/8.81 top(mark(X)) -> top(proper(X)) 24.76/8.81 top(ok(X)) -> top(active(X)) 24.76/8.81 24.76/8.81 S is empty. 24.76/8.81 Rewrite Strategy: FULL 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (9) DecreasingLoopProof (LOWER BOUND(ID)) 24.76/8.81 The following loop(s) give(s) rise to the lower bound Omega(n^1): 24.76/8.81 24.76/8.81 The rewrite sequence 24.76/8.81 24.76/8.81 sel(X1, mark(X2)) ->^+ mark(sel(X1, X2)) 24.76/8.81 24.76/8.81 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 24.76/8.81 24.76/8.81 The pumping substitution is [X2 / mark(X2)]. 24.76/8.81 24.76/8.81 The result substitution is [ ]. 24.76/8.81 24.76/8.81 24.76/8.81 24.76/8.81 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (10) 24.76/8.81 Complex Obligation (BEST) 24.76/8.81 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (11) 24.76/8.81 Obligation: 24.76/8.81 Proved the lower bound n^1 for the following obligation: 24.76/8.81 24.76/8.81 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.76/8.81 24.76/8.81 24.76/8.81 The TRS R consists of the following rules: 24.76/8.81 24.76/8.81 active(from(X)) -> mark(cons(X, from(s(X)))) 24.76/8.81 active(sel(0, cons(X, Y))) -> mark(X) 24.76/8.81 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 24.76/8.81 active(from(X)) -> from(active(X)) 24.76/8.81 active(cons(X1, X2)) -> cons(active(X1), X2) 24.76/8.81 active(s(X)) -> s(active(X)) 24.76/8.81 active(sel(X1, X2)) -> sel(active(X1), X2) 24.76/8.81 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.76/8.81 from(mark(X)) -> mark(from(X)) 24.76/8.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.76/8.81 s(mark(X)) -> mark(s(X)) 24.76/8.81 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.76/8.81 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.76/8.81 proper(from(X)) -> from(proper(X)) 24.76/8.81 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.76/8.81 proper(s(X)) -> s(proper(X)) 24.76/8.81 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.76/8.81 proper(0) -> ok(0) 24.76/8.81 from(ok(X)) -> ok(from(X)) 24.76/8.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.76/8.81 s(ok(X)) -> ok(s(X)) 24.76/8.81 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.76/8.81 top(mark(X)) -> top(proper(X)) 24.76/8.81 top(ok(X)) -> top(active(X)) 24.76/8.81 24.76/8.81 S is empty. 24.76/8.81 Rewrite Strategy: FULL 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (12) LowerBoundPropagationProof (FINISHED) 24.76/8.81 Propagated lower bound. 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (13) 24.76/8.81 BOUNDS(n^1, INF) 24.76/8.81 24.76/8.81 ---------------------------------------- 24.76/8.81 24.76/8.81 (14) 24.76/8.81 Obligation: 24.76/8.81 Analyzing the following TRS for decreasing loops: 24.76/8.81 24.76/8.81 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.76/8.81 24.76/8.81 24.76/8.81 The TRS R consists of the following rules: 24.76/8.81 24.76/8.81 active(from(X)) -> mark(cons(X, from(s(X)))) 24.76/8.81 active(sel(0, cons(X, Y))) -> mark(X) 24.76/8.81 active(sel(s(X), cons(Y, Z))) -> mark(sel(X, Z)) 24.76/8.81 active(from(X)) -> from(active(X)) 24.76/8.81 active(cons(X1, X2)) -> cons(active(X1), X2) 24.76/8.81 active(s(X)) -> s(active(X)) 24.76/8.81 active(sel(X1, X2)) -> sel(active(X1), X2) 24.76/8.81 active(sel(X1, X2)) -> sel(X1, active(X2)) 24.76/8.81 from(mark(X)) -> mark(from(X)) 24.76/8.81 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.76/8.81 s(mark(X)) -> mark(s(X)) 24.76/8.81 sel(mark(X1), X2) -> mark(sel(X1, X2)) 24.76/8.81 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 24.76/8.81 proper(from(X)) -> from(proper(X)) 24.76/8.81 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.76/8.81 proper(s(X)) -> s(proper(X)) 24.76/8.81 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 24.76/8.81 proper(0) -> ok(0) 24.76/8.81 from(ok(X)) -> ok(from(X)) 24.76/8.81 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.76/8.81 s(ok(X)) -> ok(s(X)) 24.76/8.81 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 24.76/8.81 top(mark(X)) -> top(proper(X)) 24.76/8.81 top(ok(X)) -> top(active(X)) 24.76/8.81 24.76/8.81 S is empty. 24.76/8.81 Rewrite Strategy: FULL 24.76/8.85 EOF