21.85/7.02 WORST_CASE(Omega(n^1), O(n^1)) 21.85/7.03 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 21.85/7.03 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 21.85/7.03 21.85/7.03 21.85/7.03 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.85/7.03 21.85/7.03 (0) CpxTRS 21.85/7.03 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 21.85/7.03 (2) CpxTRS 21.85/7.03 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 21.85/7.03 (4) CpxTRS 21.85/7.03 (5) CpxTrsMatchBoundsTAProof [FINISHED, 70 ms] 21.85/7.03 (6) BOUNDS(1, n^1) 21.85/7.03 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 21.85/7.03 (8) TRS for Loop Detection 21.85/7.03 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 21.85/7.03 (10) BEST 21.85/7.03 (11) proven lower bound 21.85/7.03 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 21.85/7.03 (13) BOUNDS(n^1, INF) 21.85/7.03 (14) TRS for Loop Detection 21.85/7.03 21.85/7.03 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (0) 21.85/7.03 Obligation: 21.85/7.03 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.85/7.03 21.85/7.03 21.85/7.03 The TRS R consists of the following rules: 21.85/7.03 21.85/7.03 active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y) 21.85/7.03 active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1))) 21.85/7.03 active(from(X)) -> mark(cons(X, from(s(X)))) 21.85/7.03 active(2nd(X)) -> 2nd(active(X)) 21.85/7.03 active(cons(X1, X2)) -> cons(active(X1), X2) 21.85/7.03 active(from(X)) -> from(active(X)) 21.85/7.03 active(s(X)) -> s(active(X)) 21.85/7.03 active(cons1(X1, X2)) -> cons1(active(X1), X2) 21.85/7.03 active(cons1(X1, X2)) -> cons1(X1, active(X2)) 21.85/7.03 2nd(mark(X)) -> mark(2nd(X)) 21.85/7.03 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.85/7.03 from(mark(X)) -> mark(from(X)) 21.85/7.03 s(mark(X)) -> mark(s(X)) 21.85/7.03 cons1(mark(X1), X2) -> mark(cons1(X1, X2)) 21.85/7.03 cons1(X1, mark(X2)) -> mark(cons1(X1, X2)) 21.85/7.03 proper(2nd(X)) -> 2nd(proper(X)) 21.85/7.03 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.85/7.03 proper(from(X)) -> from(proper(X)) 21.85/7.03 proper(s(X)) -> s(proper(X)) 21.85/7.03 proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2)) 21.85/7.03 2nd(ok(X)) -> ok(2nd(X)) 21.85/7.03 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.85/7.03 from(ok(X)) -> ok(from(X)) 21.85/7.03 s(ok(X)) -> ok(s(X)) 21.85/7.03 cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)) 21.85/7.03 top(mark(X)) -> top(proper(X)) 21.85/7.03 top(ok(X)) -> top(active(X)) 21.85/7.03 21.85/7.03 S is empty. 21.85/7.03 Rewrite Strategy: FULL 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 21.85/7.03 The following defined symbols can occur below the 0th argument of top: proper, active 21.85/7.03 The following defined symbols can occur below the 0th argument of proper: proper, active 21.85/7.03 The following defined symbols can occur below the 0th argument of active: proper, active 21.85/7.03 21.85/7.03 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 21.85/7.03 active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y) 21.85/7.03 active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1))) 21.85/7.03 active(from(X)) -> mark(cons(X, from(s(X)))) 21.85/7.03 active(2nd(X)) -> 2nd(active(X)) 21.85/7.03 active(cons(X1, X2)) -> cons(active(X1), X2) 21.85/7.03 active(from(X)) -> from(active(X)) 21.85/7.03 active(s(X)) -> s(active(X)) 21.85/7.03 active(cons1(X1, X2)) -> cons1(active(X1), X2) 21.85/7.03 active(cons1(X1, X2)) -> cons1(X1, active(X2)) 21.85/7.03 proper(2nd(X)) -> 2nd(proper(X)) 21.85/7.03 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.85/7.03 proper(from(X)) -> from(proper(X)) 21.85/7.03 proper(s(X)) -> s(proper(X)) 21.85/7.03 proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2)) 21.85/7.03 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (2) 21.85/7.03 Obligation: 21.85/7.03 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 21.85/7.03 21.85/7.03 21.85/7.03 The TRS R consists of the following rules: 21.85/7.03 21.85/7.03 2nd(mark(X)) -> mark(2nd(X)) 21.85/7.03 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.85/7.03 from(mark(X)) -> mark(from(X)) 21.85/7.03 s(mark(X)) -> mark(s(X)) 21.85/7.03 cons1(mark(X1), X2) -> mark(cons1(X1, X2)) 21.85/7.03 cons1(X1, mark(X2)) -> mark(cons1(X1, X2)) 21.85/7.03 2nd(ok(X)) -> ok(2nd(X)) 21.85/7.03 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.85/7.03 from(ok(X)) -> ok(from(X)) 21.85/7.03 s(ok(X)) -> ok(s(X)) 21.85/7.03 cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)) 21.85/7.03 top(mark(X)) -> top(proper(X)) 21.85/7.03 top(ok(X)) -> top(active(X)) 21.85/7.03 21.85/7.03 S is empty. 21.85/7.03 Rewrite Strategy: FULL 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 21.85/7.03 transformed relative TRS to TRS 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (4) 21.85/7.03 Obligation: 21.85/7.03 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 21.85/7.03 21.85/7.03 21.85/7.03 The TRS R consists of the following rules: 21.85/7.03 21.85/7.03 2nd(mark(X)) -> mark(2nd(X)) 21.85/7.03 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.85/7.03 from(mark(X)) -> mark(from(X)) 21.85/7.03 s(mark(X)) -> mark(s(X)) 21.85/7.03 cons1(mark(X1), X2) -> mark(cons1(X1, X2)) 21.85/7.03 cons1(X1, mark(X2)) -> mark(cons1(X1, X2)) 21.85/7.03 2nd(ok(X)) -> ok(2nd(X)) 21.85/7.03 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.85/7.03 from(ok(X)) -> ok(from(X)) 21.85/7.03 s(ok(X)) -> ok(s(X)) 21.85/7.03 cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)) 21.85/7.03 top(mark(X)) -> top(proper(X)) 21.85/7.03 top(ok(X)) -> top(active(X)) 21.85/7.03 21.85/7.03 S is empty. 21.85/7.03 Rewrite Strategy: FULL 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (5) CpxTrsMatchBoundsTAProof (FINISHED) 21.85/7.03 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. 21.85/7.03 21.85/7.03 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 21.85/7.03 final states : [1, 2, 3, 4, 5, 6] 21.85/7.03 transitions: 21.85/7.03 mark0(0) -> 0 21.85/7.03 ok0(0) -> 0 21.85/7.03 proper0(0) -> 0 21.85/7.03 active0(0) -> 0 21.85/7.03 2nd0(0) -> 1 21.85/7.03 cons0(0, 0) -> 2 21.85/7.03 from0(0) -> 3 21.85/7.03 s0(0) -> 4 21.85/7.03 cons10(0, 0) -> 5 21.85/7.03 top0(0) -> 6 21.85/7.03 2nd1(0) -> 7 21.85/7.03 mark1(7) -> 1 21.85/7.03 cons1(0, 0) -> 8 21.85/7.03 mark1(8) -> 2 21.85/7.03 from1(0) -> 9 21.85/7.03 mark1(9) -> 3 21.85/7.03 s1(0) -> 10 21.85/7.03 mark1(10) -> 4 21.85/7.03 cons11(0, 0) -> 11 21.85/7.03 mark1(11) -> 5 21.85/7.03 2nd1(0) -> 12 21.85/7.03 ok1(12) -> 1 21.85/7.03 cons1(0, 0) -> 13 21.85/7.03 ok1(13) -> 2 21.85/7.03 from1(0) -> 14 21.85/7.03 ok1(14) -> 3 21.85/7.03 s1(0) -> 15 21.85/7.03 ok1(15) -> 4 21.85/7.03 cons11(0, 0) -> 16 21.85/7.03 ok1(16) -> 5 21.85/7.03 proper1(0) -> 17 21.85/7.03 top1(17) -> 6 21.85/7.03 active1(0) -> 18 21.85/7.03 top1(18) -> 6 21.85/7.03 mark1(7) -> 7 21.85/7.03 mark1(7) -> 12 21.85/7.03 mark1(8) -> 8 21.85/7.03 mark1(8) -> 13 21.85/7.03 mark1(9) -> 9 21.85/7.03 mark1(9) -> 14 21.85/7.03 mark1(10) -> 10 21.85/7.03 mark1(10) -> 15 21.85/7.03 mark1(11) -> 11 21.85/7.03 mark1(11) -> 16 21.85/7.03 ok1(12) -> 7 21.85/7.03 ok1(12) -> 12 21.85/7.03 ok1(13) -> 8 21.85/7.03 ok1(13) -> 13 21.85/7.03 ok1(14) -> 9 21.85/7.03 ok1(14) -> 14 21.85/7.03 ok1(15) -> 10 21.85/7.03 ok1(15) -> 15 21.85/7.03 ok1(16) -> 11 21.85/7.03 ok1(16) -> 16 21.85/7.03 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (6) 21.85/7.03 BOUNDS(1, n^1) 21.85/7.03 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 21.85/7.03 Transformed a relative TRS into a decreasing-loop problem. 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (8) 21.85/7.03 Obligation: 21.85/7.03 Analyzing the following TRS for decreasing loops: 21.85/7.03 21.85/7.03 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.85/7.03 21.85/7.03 21.85/7.03 The TRS R consists of the following rules: 21.85/7.03 21.85/7.03 active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y) 21.85/7.03 active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1))) 21.85/7.03 active(from(X)) -> mark(cons(X, from(s(X)))) 21.85/7.03 active(2nd(X)) -> 2nd(active(X)) 21.85/7.03 active(cons(X1, X2)) -> cons(active(X1), X2) 21.85/7.03 active(from(X)) -> from(active(X)) 21.85/7.03 active(s(X)) -> s(active(X)) 21.85/7.03 active(cons1(X1, X2)) -> cons1(active(X1), X2) 21.85/7.03 active(cons1(X1, X2)) -> cons1(X1, active(X2)) 21.85/7.03 2nd(mark(X)) -> mark(2nd(X)) 21.85/7.03 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.85/7.03 from(mark(X)) -> mark(from(X)) 21.85/7.03 s(mark(X)) -> mark(s(X)) 21.85/7.03 cons1(mark(X1), X2) -> mark(cons1(X1, X2)) 21.85/7.03 cons1(X1, mark(X2)) -> mark(cons1(X1, X2)) 21.85/7.03 proper(2nd(X)) -> 2nd(proper(X)) 21.85/7.03 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.85/7.03 proper(from(X)) -> from(proper(X)) 21.85/7.03 proper(s(X)) -> s(proper(X)) 21.85/7.03 proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2)) 21.85/7.03 2nd(ok(X)) -> ok(2nd(X)) 21.85/7.03 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.85/7.03 from(ok(X)) -> ok(from(X)) 21.85/7.03 s(ok(X)) -> ok(s(X)) 21.85/7.03 cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)) 21.85/7.03 top(mark(X)) -> top(proper(X)) 21.85/7.03 top(ok(X)) -> top(active(X)) 21.85/7.03 21.85/7.03 S is empty. 21.85/7.03 Rewrite Strategy: FULL 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (9) DecreasingLoopProof (LOWER BOUND(ID)) 21.85/7.03 The following loop(s) give(s) rise to the lower bound Omega(n^1): 21.85/7.03 21.85/7.03 The rewrite sequence 21.85/7.03 21.85/7.03 2nd(ok(X)) ->^+ ok(2nd(X)) 21.85/7.03 21.85/7.03 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 21.85/7.03 21.85/7.03 The pumping substitution is [X / ok(X)]. 21.85/7.03 21.85/7.03 The result substitution is [ ]. 21.85/7.03 21.85/7.03 21.85/7.03 21.85/7.03 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (10) 21.85/7.03 Complex Obligation (BEST) 21.85/7.03 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (11) 21.85/7.03 Obligation: 21.85/7.03 Proved the lower bound n^1 for the following obligation: 21.85/7.03 21.85/7.03 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.85/7.03 21.85/7.03 21.85/7.03 The TRS R consists of the following rules: 21.85/7.03 21.85/7.03 active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y) 21.85/7.03 active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1))) 21.85/7.03 active(from(X)) -> mark(cons(X, from(s(X)))) 21.85/7.03 active(2nd(X)) -> 2nd(active(X)) 21.85/7.03 active(cons(X1, X2)) -> cons(active(X1), X2) 21.85/7.03 active(from(X)) -> from(active(X)) 21.85/7.03 active(s(X)) -> s(active(X)) 21.85/7.03 active(cons1(X1, X2)) -> cons1(active(X1), X2) 21.85/7.03 active(cons1(X1, X2)) -> cons1(X1, active(X2)) 21.85/7.03 2nd(mark(X)) -> mark(2nd(X)) 21.85/7.03 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.85/7.03 from(mark(X)) -> mark(from(X)) 21.85/7.03 s(mark(X)) -> mark(s(X)) 21.85/7.03 cons1(mark(X1), X2) -> mark(cons1(X1, X2)) 21.85/7.03 cons1(X1, mark(X2)) -> mark(cons1(X1, X2)) 21.85/7.03 proper(2nd(X)) -> 2nd(proper(X)) 21.85/7.03 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.85/7.03 proper(from(X)) -> from(proper(X)) 21.85/7.03 proper(s(X)) -> s(proper(X)) 21.85/7.03 proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2)) 21.85/7.03 2nd(ok(X)) -> ok(2nd(X)) 21.85/7.03 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.85/7.03 from(ok(X)) -> ok(from(X)) 21.85/7.03 s(ok(X)) -> ok(s(X)) 21.85/7.03 cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)) 21.85/7.03 top(mark(X)) -> top(proper(X)) 21.85/7.03 top(ok(X)) -> top(active(X)) 21.85/7.03 21.85/7.03 S is empty. 21.85/7.03 Rewrite Strategy: FULL 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (12) LowerBoundPropagationProof (FINISHED) 21.85/7.03 Propagated lower bound. 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (13) 21.85/7.03 BOUNDS(n^1, INF) 21.85/7.03 21.85/7.03 ---------------------------------------- 21.85/7.03 21.85/7.03 (14) 21.85/7.03 Obligation: 21.85/7.03 Analyzing the following TRS for decreasing loops: 21.85/7.03 21.85/7.03 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.85/7.03 21.85/7.03 21.85/7.03 The TRS R consists of the following rules: 21.85/7.03 21.85/7.03 active(2nd(cons1(X, cons(Y, Z)))) -> mark(Y) 21.85/7.03 active(2nd(cons(X, X1))) -> mark(2nd(cons1(X, X1))) 21.85/7.03 active(from(X)) -> mark(cons(X, from(s(X)))) 21.85/7.03 active(2nd(X)) -> 2nd(active(X)) 21.85/7.03 active(cons(X1, X2)) -> cons(active(X1), X2) 21.85/7.03 active(from(X)) -> from(active(X)) 21.85/7.03 active(s(X)) -> s(active(X)) 21.85/7.03 active(cons1(X1, X2)) -> cons1(active(X1), X2) 21.85/7.03 active(cons1(X1, X2)) -> cons1(X1, active(X2)) 21.85/7.03 2nd(mark(X)) -> mark(2nd(X)) 21.85/7.03 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.85/7.03 from(mark(X)) -> mark(from(X)) 21.85/7.03 s(mark(X)) -> mark(s(X)) 21.85/7.03 cons1(mark(X1), X2) -> mark(cons1(X1, X2)) 21.85/7.03 cons1(X1, mark(X2)) -> mark(cons1(X1, X2)) 21.85/7.03 proper(2nd(X)) -> 2nd(proper(X)) 21.85/7.03 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.85/7.03 proper(from(X)) -> from(proper(X)) 21.85/7.03 proper(s(X)) -> s(proper(X)) 21.85/7.03 proper(cons1(X1, X2)) -> cons1(proper(X1), proper(X2)) 21.85/7.03 2nd(ok(X)) -> ok(2nd(X)) 21.85/7.03 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.85/7.03 from(ok(X)) -> ok(from(X)) 21.85/7.03 s(ok(X)) -> ok(s(X)) 21.85/7.03 cons1(ok(X1), ok(X2)) -> ok(cons1(X1, X2)) 21.85/7.03 top(mark(X)) -> top(proper(X)) 21.85/7.03 top(ok(X)) -> top(active(X)) 21.85/7.03 21.85/7.03 S is empty. 21.85/7.03 Rewrite Strategy: FULL 21.94/7.07 EOF