21.28/7.82 WORST_CASE(Omega(n^1), O(n^1)) 21.28/7.83 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 21.28/7.83 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 21.28/7.83 21.28/7.83 21.28/7.83 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.28/7.83 21.28/7.83 (0) CpxTRS 21.28/7.83 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 21.28/7.83 (2) CpxTRS 21.28/7.83 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 21.28/7.83 (4) CpxTRS 21.28/7.83 (5) CpxTrsMatchBoundsTAProof [FINISHED, 69 ms] 21.28/7.83 (6) BOUNDS(1, n^1) 21.28/7.83 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 21.28/7.83 (8) TRS for Loop Detection 21.28/7.83 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 21.28/7.83 (10) BEST 21.28/7.83 (11) proven lower bound 21.28/7.83 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 21.28/7.83 (13) BOUNDS(n^1, INF) 21.28/7.83 (14) TRS for Loop Detection 21.28/7.83 21.28/7.83 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (0) 21.28/7.83 Obligation: 21.28/7.83 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.28/7.83 21.28/7.83 21.28/7.83 The TRS R consists of the following rules: 21.28/7.83 21.28/7.83 active(from(X)) -> mark(cons(X, from(s(X)))) 21.28/7.83 active(length(nil)) -> mark(0) 21.28/7.83 active(length(cons(X, Y))) -> mark(s(length1(Y))) 21.28/7.83 active(length1(X)) -> mark(length(X)) 21.28/7.83 active(from(X)) -> from(active(X)) 21.28/7.83 active(cons(X1, X2)) -> cons(active(X1), X2) 21.28/7.83 active(s(X)) -> s(active(X)) 21.28/7.83 from(mark(X)) -> mark(from(X)) 21.28/7.83 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.28/7.83 s(mark(X)) -> mark(s(X)) 21.28/7.83 proper(from(X)) -> from(proper(X)) 21.28/7.83 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.28/7.83 proper(s(X)) -> s(proper(X)) 21.28/7.83 proper(length(X)) -> length(proper(X)) 21.28/7.83 proper(nil) -> ok(nil) 21.28/7.83 proper(0) -> ok(0) 21.28/7.83 proper(length1(X)) -> length1(proper(X)) 21.28/7.83 from(ok(X)) -> ok(from(X)) 21.28/7.83 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.28/7.83 s(ok(X)) -> ok(s(X)) 21.28/7.83 length(ok(X)) -> ok(length(X)) 21.28/7.83 length1(ok(X)) -> ok(length1(X)) 21.28/7.83 top(mark(X)) -> top(proper(X)) 21.28/7.83 top(ok(X)) -> top(active(X)) 21.28/7.83 21.28/7.83 S is empty. 21.28/7.83 Rewrite Strategy: FULL 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 21.28/7.83 The following defined symbols can occur below the 0th argument of top: proper, active 21.28/7.83 The following defined symbols can occur below the 0th argument of proper: proper, active 21.28/7.83 The following defined symbols can occur below the 0th argument of active: proper, active 21.28/7.83 21.28/7.83 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 21.28/7.83 active(from(X)) -> mark(cons(X, from(s(X)))) 21.28/7.83 active(length(nil)) -> mark(0) 21.28/7.83 active(length(cons(X, Y))) -> mark(s(length1(Y))) 21.28/7.83 active(length1(X)) -> mark(length(X)) 21.28/7.83 active(from(X)) -> from(active(X)) 21.28/7.83 active(cons(X1, X2)) -> cons(active(X1), X2) 21.28/7.83 active(s(X)) -> s(active(X)) 21.28/7.83 proper(from(X)) -> from(proper(X)) 21.28/7.83 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.28/7.83 proper(s(X)) -> s(proper(X)) 21.28/7.83 proper(length(X)) -> length(proper(X)) 21.28/7.83 proper(length1(X)) -> length1(proper(X)) 21.28/7.83 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (2) 21.28/7.83 Obligation: 21.28/7.83 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 21.28/7.83 21.28/7.83 21.28/7.83 The TRS R consists of the following rules: 21.28/7.83 21.28/7.83 from(mark(X)) -> mark(from(X)) 21.28/7.83 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.28/7.83 s(mark(X)) -> mark(s(X)) 21.28/7.83 proper(nil) -> ok(nil) 21.28/7.83 proper(0) -> ok(0) 21.28/7.83 from(ok(X)) -> ok(from(X)) 21.28/7.83 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.28/7.83 s(ok(X)) -> ok(s(X)) 21.28/7.83 length(ok(X)) -> ok(length(X)) 21.28/7.83 length1(ok(X)) -> ok(length1(X)) 21.28/7.83 top(mark(X)) -> top(proper(X)) 21.28/7.83 top(ok(X)) -> top(active(X)) 21.28/7.83 21.28/7.83 S is empty. 21.28/7.83 Rewrite Strategy: FULL 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 21.28/7.83 transformed relative TRS to TRS 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (4) 21.28/7.83 Obligation: 21.28/7.83 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 21.28/7.83 21.28/7.83 21.28/7.83 The TRS R consists of the following rules: 21.28/7.83 21.28/7.83 from(mark(X)) -> mark(from(X)) 21.28/7.83 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.28/7.83 s(mark(X)) -> mark(s(X)) 21.28/7.83 proper(nil) -> ok(nil) 21.28/7.83 proper(0) -> ok(0) 21.28/7.83 from(ok(X)) -> ok(from(X)) 21.28/7.83 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.28/7.83 s(ok(X)) -> ok(s(X)) 21.28/7.83 length(ok(X)) -> ok(length(X)) 21.28/7.83 length1(ok(X)) -> ok(length1(X)) 21.28/7.83 top(mark(X)) -> top(proper(X)) 21.28/7.83 top(ok(X)) -> top(active(X)) 21.28/7.83 21.28/7.83 S is empty. 21.28/7.83 Rewrite Strategy: FULL 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (5) CpxTrsMatchBoundsTAProof (FINISHED) 21.28/7.83 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. 21.28/7.83 21.28/7.83 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 21.28/7.83 final states : [1, 2, 3, 4, 5, 6, 7] 21.28/7.83 transitions: 21.28/7.83 mark0(0) -> 0 21.28/7.83 nil0() -> 0 21.28/7.83 ok0(0) -> 0 21.28/7.83 00() -> 0 21.28/7.83 active0(0) -> 0 21.28/7.83 from0(0) -> 1 21.28/7.83 cons0(0, 0) -> 2 21.28/7.83 s0(0) -> 3 21.28/7.83 proper0(0) -> 4 21.28/7.83 length0(0) -> 5 21.28/7.83 length10(0) -> 6 21.28/7.83 top0(0) -> 7 21.28/7.83 from1(0) -> 8 21.28/7.83 mark1(8) -> 1 21.28/7.83 cons1(0, 0) -> 9 21.28/7.83 mark1(9) -> 2 21.28/7.83 s1(0) -> 10 21.28/7.83 mark1(10) -> 3 21.28/7.83 nil1() -> 11 21.28/7.83 ok1(11) -> 4 21.28/7.83 01() -> 12 21.28/7.83 ok1(12) -> 4 21.28/7.83 from1(0) -> 13 21.28/7.83 ok1(13) -> 1 21.28/7.83 cons1(0, 0) -> 14 21.28/7.83 ok1(14) -> 2 21.28/7.83 s1(0) -> 15 21.28/7.83 ok1(15) -> 3 21.28/7.83 length1(0) -> 16 21.28/7.83 ok1(16) -> 5 21.28/7.83 length11(0) -> 17 21.28/7.83 ok1(17) -> 6 21.28/7.83 proper1(0) -> 18 21.28/7.83 top1(18) -> 7 21.28/7.83 active1(0) -> 19 21.28/7.83 top1(19) -> 7 21.28/7.83 mark1(8) -> 8 21.28/7.83 mark1(8) -> 13 21.28/7.83 mark1(9) -> 9 21.28/7.83 mark1(9) -> 14 21.28/7.83 mark1(10) -> 10 21.28/7.83 mark1(10) -> 15 21.28/7.83 ok1(11) -> 18 21.28/7.83 ok1(12) -> 18 21.28/7.83 ok1(13) -> 8 21.28/7.83 ok1(13) -> 13 21.28/7.83 ok1(14) -> 9 21.28/7.83 ok1(14) -> 14 21.28/7.83 ok1(15) -> 10 21.28/7.83 ok1(15) -> 15 21.28/7.83 ok1(16) -> 16 21.28/7.83 ok1(17) -> 17 21.28/7.83 active2(11) -> 20 21.28/7.83 top2(20) -> 7 21.28/7.83 active2(12) -> 20 21.28/7.83 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (6) 21.28/7.83 BOUNDS(1, n^1) 21.28/7.83 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 21.28/7.83 Transformed a relative TRS into a decreasing-loop problem. 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (8) 21.28/7.83 Obligation: 21.28/7.83 Analyzing the following TRS for decreasing loops: 21.28/7.83 21.28/7.83 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.28/7.83 21.28/7.83 21.28/7.83 The TRS R consists of the following rules: 21.28/7.83 21.28/7.83 active(from(X)) -> mark(cons(X, from(s(X)))) 21.28/7.83 active(length(nil)) -> mark(0) 21.28/7.83 active(length(cons(X, Y))) -> mark(s(length1(Y))) 21.28/7.83 active(length1(X)) -> mark(length(X)) 21.28/7.83 active(from(X)) -> from(active(X)) 21.28/7.83 active(cons(X1, X2)) -> cons(active(X1), X2) 21.28/7.83 active(s(X)) -> s(active(X)) 21.28/7.83 from(mark(X)) -> mark(from(X)) 21.28/7.83 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.28/7.83 s(mark(X)) -> mark(s(X)) 21.28/7.83 proper(from(X)) -> from(proper(X)) 21.28/7.83 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.28/7.83 proper(s(X)) -> s(proper(X)) 21.28/7.83 proper(length(X)) -> length(proper(X)) 21.28/7.83 proper(nil) -> ok(nil) 21.28/7.83 proper(0) -> ok(0) 21.28/7.83 proper(length1(X)) -> length1(proper(X)) 21.28/7.83 from(ok(X)) -> ok(from(X)) 21.28/7.83 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.28/7.83 s(ok(X)) -> ok(s(X)) 21.28/7.83 length(ok(X)) -> ok(length(X)) 21.28/7.83 length1(ok(X)) -> ok(length1(X)) 21.28/7.83 top(mark(X)) -> top(proper(X)) 21.28/7.83 top(ok(X)) -> top(active(X)) 21.28/7.83 21.28/7.83 S is empty. 21.28/7.83 Rewrite Strategy: FULL 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (9) DecreasingLoopProof (LOWER BOUND(ID)) 21.28/7.83 The following loop(s) give(s) rise to the lower bound Omega(n^1): 21.28/7.83 21.28/7.83 The rewrite sequence 21.28/7.83 21.28/7.83 s(mark(X)) ->^+ mark(s(X)) 21.28/7.83 21.28/7.83 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 21.28/7.83 21.28/7.83 The pumping substitution is [X / mark(X)]. 21.28/7.83 21.28/7.83 The result substitution is [ ]. 21.28/7.83 21.28/7.83 21.28/7.83 21.28/7.83 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (10) 21.28/7.83 Complex Obligation (BEST) 21.28/7.83 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (11) 21.28/7.83 Obligation: 21.28/7.83 Proved the lower bound n^1 for the following obligation: 21.28/7.83 21.28/7.83 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.28/7.83 21.28/7.83 21.28/7.83 The TRS R consists of the following rules: 21.28/7.83 21.28/7.83 active(from(X)) -> mark(cons(X, from(s(X)))) 21.28/7.83 active(length(nil)) -> mark(0) 21.28/7.83 active(length(cons(X, Y))) -> mark(s(length1(Y))) 21.28/7.83 active(length1(X)) -> mark(length(X)) 21.28/7.83 active(from(X)) -> from(active(X)) 21.28/7.83 active(cons(X1, X2)) -> cons(active(X1), X2) 21.28/7.83 active(s(X)) -> s(active(X)) 21.28/7.83 from(mark(X)) -> mark(from(X)) 21.28/7.83 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.28/7.83 s(mark(X)) -> mark(s(X)) 21.28/7.83 proper(from(X)) -> from(proper(X)) 21.28/7.83 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.28/7.83 proper(s(X)) -> s(proper(X)) 21.28/7.83 proper(length(X)) -> length(proper(X)) 21.28/7.83 proper(nil) -> ok(nil) 21.28/7.83 proper(0) -> ok(0) 21.28/7.83 proper(length1(X)) -> length1(proper(X)) 21.28/7.83 from(ok(X)) -> ok(from(X)) 21.28/7.83 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.28/7.83 s(ok(X)) -> ok(s(X)) 21.28/7.83 length(ok(X)) -> ok(length(X)) 21.28/7.83 length1(ok(X)) -> ok(length1(X)) 21.28/7.83 top(mark(X)) -> top(proper(X)) 21.28/7.83 top(ok(X)) -> top(active(X)) 21.28/7.83 21.28/7.83 S is empty. 21.28/7.83 Rewrite Strategy: FULL 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (12) LowerBoundPropagationProof (FINISHED) 21.28/7.83 Propagated lower bound. 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (13) 21.28/7.83 BOUNDS(n^1, INF) 21.28/7.83 21.28/7.83 ---------------------------------------- 21.28/7.83 21.28/7.83 (14) 21.28/7.83 Obligation: 21.28/7.83 Analyzing the following TRS for decreasing loops: 21.28/7.83 21.28/7.83 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 21.28/7.83 21.28/7.83 21.28/7.83 The TRS R consists of the following rules: 21.28/7.83 21.28/7.83 active(from(X)) -> mark(cons(X, from(s(X)))) 21.28/7.83 active(length(nil)) -> mark(0) 21.28/7.83 active(length(cons(X, Y))) -> mark(s(length1(Y))) 21.28/7.83 active(length1(X)) -> mark(length(X)) 21.28/7.83 active(from(X)) -> from(active(X)) 21.28/7.83 active(cons(X1, X2)) -> cons(active(X1), X2) 21.28/7.83 active(s(X)) -> s(active(X)) 21.28/7.83 from(mark(X)) -> mark(from(X)) 21.28/7.83 cons(mark(X1), X2) -> mark(cons(X1, X2)) 21.28/7.83 s(mark(X)) -> mark(s(X)) 21.28/7.83 proper(from(X)) -> from(proper(X)) 21.28/7.83 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 21.28/7.83 proper(s(X)) -> s(proper(X)) 21.28/7.83 proper(length(X)) -> length(proper(X)) 21.28/7.83 proper(nil) -> ok(nil) 21.28/7.83 proper(0) -> ok(0) 21.28/7.83 proper(length1(X)) -> length1(proper(X)) 21.28/7.83 from(ok(X)) -> ok(from(X)) 21.28/7.83 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 21.28/7.83 s(ok(X)) -> ok(s(X)) 21.28/7.83 length(ok(X)) -> ok(length(X)) 21.28/7.83 length1(ok(X)) -> ok(length1(X)) 21.28/7.83 top(mark(X)) -> top(proper(X)) 21.28/7.83 top(ok(X)) -> top(active(X)) 21.28/7.83 21.28/7.83 S is empty. 21.28/7.83 Rewrite Strategy: FULL 21.60/9.56 EOF