20.87/7.40 WORST_CASE(Omega(n^1), O(n^1)) 20.87/7.41 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 20.87/7.41 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 20.87/7.41 20.87/7.41 20.87/7.41 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.87/7.41 20.87/7.41 (0) CpxTRS 20.87/7.41 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 20.87/7.41 (2) CpxTRS 20.87/7.41 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 20.87/7.41 (4) CpxTRS 20.87/7.41 (5) CpxTrsMatchBoundsTAProof [FINISHED, 41 ms] 20.87/7.41 (6) BOUNDS(1, n^1) 20.87/7.41 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 20.87/7.41 (8) TRS for Loop Detection 20.87/7.41 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 20.87/7.41 (10) BEST 20.87/7.41 (11) proven lower bound 20.87/7.41 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 20.87/7.41 (13) BOUNDS(n^1, INF) 20.87/7.41 (14) TRS for Loop Detection 20.87/7.41 20.87/7.41 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (0) 20.87/7.41 Obligation: 20.87/7.41 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.87/7.41 20.87/7.41 20.87/7.41 The TRS R consists of the following rules: 20.87/7.41 20.87/7.41 active(2nd(cons(X, cons(Y, Z)))) -> mark(Y) 20.87/7.41 active(from(X)) -> mark(cons(X, from(s(X)))) 20.87/7.41 active(2nd(X)) -> 2nd(active(X)) 20.87/7.41 active(cons(X1, X2)) -> cons(active(X1), X2) 20.87/7.41 active(from(X)) -> from(active(X)) 20.87/7.41 active(s(X)) -> s(active(X)) 20.87/7.41 2nd(mark(X)) -> mark(2nd(X)) 20.87/7.41 cons(mark(X1), X2) -> mark(cons(X1, X2)) 20.87/7.41 from(mark(X)) -> mark(from(X)) 20.87/7.41 s(mark(X)) -> mark(s(X)) 20.87/7.41 proper(2nd(X)) -> 2nd(proper(X)) 20.87/7.41 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 20.87/7.41 proper(from(X)) -> from(proper(X)) 20.87/7.41 proper(s(X)) -> s(proper(X)) 20.87/7.41 2nd(ok(X)) -> ok(2nd(X)) 20.87/7.41 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 20.87/7.41 from(ok(X)) -> ok(from(X)) 20.87/7.41 s(ok(X)) -> ok(s(X)) 20.87/7.41 top(mark(X)) -> top(proper(X)) 20.87/7.41 top(ok(X)) -> top(active(X)) 20.87/7.41 20.87/7.41 S is empty. 20.87/7.41 Rewrite Strategy: FULL 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 20.87/7.41 The following defined symbols can occur below the 0th argument of top: proper, active 20.87/7.41 The following defined symbols can occur below the 0th argument of proper: proper, active 20.87/7.41 The following defined symbols can occur below the 0th argument of active: proper, active 20.87/7.41 20.87/7.41 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 20.87/7.41 active(2nd(cons(X, cons(Y, Z)))) -> mark(Y) 20.87/7.41 active(from(X)) -> mark(cons(X, from(s(X)))) 20.87/7.41 active(2nd(X)) -> 2nd(active(X)) 20.87/7.41 active(cons(X1, X2)) -> cons(active(X1), X2) 20.87/7.41 active(from(X)) -> from(active(X)) 20.87/7.41 active(s(X)) -> s(active(X)) 20.87/7.41 proper(2nd(X)) -> 2nd(proper(X)) 20.87/7.41 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 20.87/7.41 proper(from(X)) -> from(proper(X)) 20.87/7.41 proper(s(X)) -> s(proper(X)) 20.87/7.41 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (2) 20.87/7.41 Obligation: 20.87/7.41 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 20.87/7.41 20.87/7.41 20.87/7.41 The TRS R consists of the following rules: 20.87/7.41 20.87/7.41 2nd(mark(X)) -> mark(2nd(X)) 20.87/7.41 cons(mark(X1), X2) -> mark(cons(X1, X2)) 20.87/7.41 from(mark(X)) -> mark(from(X)) 20.87/7.41 s(mark(X)) -> mark(s(X)) 20.87/7.41 2nd(ok(X)) -> ok(2nd(X)) 20.87/7.41 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 20.87/7.41 from(ok(X)) -> ok(from(X)) 20.87/7.41 s(ok(X)) -> ok(s(X)) 20.87/7.41 top(mark(X)) -> top(proper(X)) 20.87/7.41 top(ok(X)) -> top(active(X)) 20.87/7.41 20.87/7.41 S is empty. 20.87/7.41 Rewrite Strategy: FULL 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 20.87/7.41 transformed relative TRS to TRS 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (4) 20.87/7.41 Obligation: 20.87/7.41 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 20.87/7.41 20.87/7.41 20.87/7.41 The TRS R consists of the following rules: 20.87/7.41 20.87/7.41 2nd(mark(X)) -> mark(2nd(X)) 20.87/7.41 cons(mark(X1), X2) -> mark(cons(X1, X2)) 20.87/7.41 from(mark(X)) -> mark(from(X)) 20.87/7.41 s(mark(X)) -> mark(s(X)) 20.87/7.41 2nd(ok(X)) -> ok(2nd(X)) 20.87/7.41 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 20.87/7.41 from(ok(X)) -> ok(from(X)) 20.87/7.41 s(ok(X)) -> ok(s(X)) 20.87/7.41 top(mark(X)) -> top(proper(X)) 20.87/7.41 top(ok(X)) -> top(active(X)) 20.87/7.41 20.87/7.41 S is empty. 20.87/7.41 Rewrite Strategy: FULL 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (5) CpxTrsMatchBoundsTAProof (FINISHED) 20.87/7.41 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 1. 20.87/7.41 20.87/7.41 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 20.87/7.41 final states : [1, 2, 3, 4, 5] 20.87/7.41 transitions: 20.87/7.41 mark0(0) -> 0 20.87/7.41 ok0(0) -> 0 20.87/7.41 proper0(0) -> 0 20.87/7.41 active0(0) -> 0 20.87/7.41 2nd0(0) -> 1 20.87/7.41 cons0(0, 0) -> 2 20.87/7.41 from0(0) -> 3 20.87/7.41 s0(0) -> 4 20.87/7.41 top0(0) -> 5 20.87/7.41 2nd1(0) -> 6 20.87/7.41 mark1(6) -> 1 20.87/7.41 cons1(0, 0) -> 7 20.87/7.41 mark1(7) -> 2 20.87/7.41 from1(0) -> 8 20.87/7.41 mark1(8) -> 3 20.87/7.41 s1(0) -> 9 20.87/7.41 mark1(9) -> 4 20.87/7.41 2nd1(0) -> 10 20.87/7.41 ok1(10) -> 1 20.87/7.41 cons1(0, 0) -> 11 20.87/7.41 ok1(11) -> 2 20.87/7.41 from1(0) -> 12 20.87/7.41 ok1(12) -> 3 20.87/7.41 s1(0) -> 13 20.87/7.41 ok1(13) -> 4 20.87/7.41 proper1(0) -> 14 20.87/7.41 top1(14) -> 5 20.87/7.41 active1(0) -> 15 20.87/7.41 top1(15) -> 5 20.87/7.41 mark1(6) -> 6 20.87/7.41 mark1(6) -> 10 20.87/7.41 mark1(7) -> 7 20.87/7.41 mark1(7) -> 11 20.87/7.41 mark1(8) -> 8 20.87/7.41 mark1(8) -> 12 20.87/7.41 mark1(9) -> 9 20.87/7.41 mark1(9) -> 13 20.87/7.41 ok1(10) -> 6 20.87/7.41 ok1(10) -> 10 20.87/7.41 ok1(11) -> 7 20.87/7.41 ok1(11) -> 11 20.87/7.41 ok1(12) -> 8 20.87/7.41 ok1(12) -> 12 20.87/7.41 ok1(13) -> 9 20.87/7.41 ok1(13) -> 13 20.87/7.41 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (6) 20.87/7.41 BOUNDS(1, n^1) 20.87/7.41 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 20.87/7.41 Transformed a relative TRS into a decreasing-loop problem. 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (8) 20.87/7.41 Obligation: 20.87/7.41 Analyzing the following TRS for decreasing loops: 20.87/7.41 20.87/7.41 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.87/7.41 20.87/7.41 20.87/7.41 The TRS R consists of the following rules: 20.87/7.41 20.87/7.41 active(2nd(cons(X, cons(Y, Z)))) -> mark(Y) 20.87/7.41 active(from(X)) -> mark(cons(X, from(s(X)))) 20.87/7.41 active(2nd(X)) -> 2nd(active(X)) 20.87/7.41 active(cons(X1, X2)) -> cons(active(X1), X2) 20.87/7.41 active(from(X)) -> from(active(X)) 20.87/7.41 active(s(X)) -> s(active(X)) 20.87/7.41 2nd(mark(X)) -> mark(2nd(X)) 20.87/7.41 cons(mark(X1), X2) -> mark(cons(X1, X2)) 20.87/7.41 from(mark(X)) -> mark(from(X)) 20.87/7.41 s(mark(X)) -> mark(s(X)) 20.87/7.41 proper(2nd(X)) -> 2nd(proper(X)) 20.87/7.41 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 20.87/7.41 proper(from(X)) -> from(proper(X)) 20.87/7.41 proper(s(X)) -> s(proper(X)) 20.87/7.41 2nd(ok(X)) -> ok(2nd(X)) 20.87/7.41 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 20.87/7.41 from(ok(X)) -> ok(from(X)) 20.87/7.41 s(ok(X)) -> ok(s(X)) 20.87/7.41 top(mark(X)) -> top(proper(X)) 20.87/7.41 top(ok(X)) -> top(active(X)) 20.87/7.41 20.87/7.41 S is empty. 20.87/7.41 Rewrite Strategy: FULL 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (9) DecreasingLoopProof (LOWER BOUND(ID)) 20.87/7.41 The following loop(s) give(s) rise to the lower bound Omega(n^1): 20.87/7.41 20.87/7.41 The rewrite sequence 20.87/7.41 20.87/7.41 2nd(ok(X)) ->^+ ok(2nd(X)) 20.87/7.41 20.87/7.41 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 20.87/7.41 20.87/7.41 The pumping substitution is [X / ok(X)]. 20.87/7.41 20.87/7.41 The result substitution is [ ]. 20.87/7.41 20.87/7.41 20.87/7.41 20.87/7.41 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (10) 20.87/7.41 Complex Obligation (BEST) 20.87/7.41 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (11) 20.87/7.41 Obligation: 20.87/7.41 Proved the lower bound n^1 for the following obligation: 20.87/7.41 20.87/7.41 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.87/7.41 20.87/7.41 20.87/7.41 The TRS R consists of the following rules: 20.87/7.41 20.87/7.41 active(2nd(cons(X, cons(Y, Z)))) -> mark(Y) 20.87/7.41 active(from(X)) -> mark(cons(X, from(s(X)))) 20.87/7.41 active(2nd(X)) -> 2nd(active(X)) 20.87/7.41 active(cons(X1, X2)) -> cons(active(X1), X2) 20.87/7.41 active(from(X)) -> from(active(X)) 20.87/7.41 active(s(X)) -> s(active(X)) 20.87/7.41 2nd(mark(X)) -> mark(2nd(X)) 20.87/7.41 cons(mark(X1), X2) -> mark(cons(X1, X2)) 20.87/7.41 from(mark(X)) -> mark(from(X)) 20.87/7.41 s(mark(X)) -> mark(s(X)) 20.87/7.41 proper(2nd(X)) -> 2nd(proper(X)) 20.87/7.41 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 20.87/7.41 proper(from(X)) -> from(proper(X)) 20.87/7.41 proper(s(X)) -> s(proper(X)) 20.87/7.41 2nd(ok(X)) -> ok(2nd(X)) 20.87/7.41 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 20.87/7.41 from(ok(X)) -> ok(from(X)) 20.87/7.41 s(ok(X)) -> ok(s(X)) 20.87/7.41 top(mark(X)) -> top(proper(X)) 20.87/7.41 top(ok(X)) -> top(active(X)) 20.87/7.41 20.87/7.41 S is empty. 20.87/7.41 Rewrite Strategy: FULL 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (12) LowerBoundPropagationProof (FINISHED) 20.87/7.41 Propagated lower bound. 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (13) 20.87/7.41 BOUNDS(n^1, INF) 20.87/7.41 20.87/7.41 ---------------------------------------- 20.87/7.41 20.87/7.41 (14) 20.87/7.41 Obligation: 20.87/7.41 Analyzing the following TRS for decreasing loops: 20.87/7.41 20.87/7.41 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 20.87/7.41 20.87/7.41 20.87/7.41 The TRS R consists of the following rules: 20.87/7.41 20.87/7.41 active(2nd(cons(X, cons(Y, Z)))) -> mark(Y) 20.87/7.41 active(from(X)) -> mark(cons(X, from(s(X)))) 20.87/7.41 active(2nd(X)) -> 2nd(active(X)) 20.87/7.41 active(cons(X1, X2)) -> cons(active(X1), X2) 20.87/7.41 active(from(X)) -> from(active(X)) 20.87/7.41 active(s(X)) -> s(active(X)) 20.87/7.41 2nd(mark(X)) -> mark(2nd(X)) 20.87/7.41 cons(mark(X1), X2) -> mark(cons(X1, X2)) 20.87/7.41 from(mark(X)) -> mark(from(X)) 20.87/7.41 s(mark(X)) -> mark(s(X)) 20.87/7.41 proper(2nd(X)) -> 2nd(proper(X)) 20.87/7.41 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 20.87/7.41 proper(from(X)) -> from(proper(X)) 20.87/7.41 proper(s(X)) -> s(proper(X)) 20.87/7.41 2nd(ok(X)) -> ok(2nd(X)) 20.87/7.41 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 20.87/7.41 from(ok(X)) -> ok(from(X)) 20.87/7.41 s(ok(X)) -> ok(s(X)) 20.87/7.41 top(mark(X)) -> top(proper(X)) 20.87/7.41 top(ok(X)) -> top(active(X)) 20.87/7.41 20.87/7.41 S is empty. 20.87/7.41 Rewrite Strategy: FULL 21.01/7.51 EOF