24.81/8.96 WORST_CASE(Omega(n^1), O(n^1)) 24.81/8.97 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 24.81/8.97 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 24.81/8.97 24.81/8.97 24.81/8.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.81/8.97 24.81/8.97 (0) CpxTRS 24.81/8.97 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 24.81/8.97 (2) CpxTRS 24.81/8.97 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 24.81/8.97 (4) CpxTRS 24.81/8.97 (5) CpxTrsMatchBoundsTAProof [FINISHED, 62 ms] 24.81/8.97 (6) BOUNDS(1, n^1) 24.81/8.97 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 24.81/8.97 (8) TRS for Loop Detection 24.81/8.97 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 24.81/8.97 (10) BEST 24.81/8.97 (11) proven lower bound 24.81/8.97 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 24.81/8.97 (13) BOUNDS(n^1, INF) 24.81/8.97 (14) TRS for Loop Detection 24.81/8.97 24.81/8.97 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (0) 24.81/8.97 Obligation: 24.81/8.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.81/8.97 24.81/8.97 24.81/8.97 The TRS R consists of the following rules: 24.81/8.97 24.81/8.97 active(first(0, X)) -> mark(nil) 24.81/8.97 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 24.81/8.97 active(from(X)) -> mark(cons(X, from(s(X)))) 24.81/8.97 active(first(X1, X2)) -> first(active(X1), X2) 24.81/8.97 active(first(X1, X2)) -> first(X1, active(X2)) 24.81/8.97 active(s(X)) -> s(active(X)) 24.81/8.97 active(cons(X1, X2)) -> cons(active(X1), X2) 24.81/8.97 active(from(X)) -> from(active(X)) 24.81/8.97 first(mark(X1), X2) -> mark(first(X1, X2)) 24.81/8.97 first(X1, mark(X2)) -> mark(first(X1, X2)) 24.81/8.97 s(mark(X)) -> mark(s(X)) 24.81/8.97 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.81/8.97 from(mark(X)) -> mark(from(X)) 24.81/8.97 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 24.81/8.97 proper(0) -> ok(0) 24.81/8.97 proper(nil) -> ok(nil) 24.81/8.97 proper(s(X)) -> s(proper(X)) 24.81/8.97 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.81/8.97 proper(from(X)) -> from(proper(X)) 24.81/8.97 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 24.81/8.97 s(ok(X)) -> ok(s(X)) 24.81/8.97 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.81/8.97 from(ok(X)) -> ok(from(X)) 24.81/8.97 top(mark(X)) -> top(proper(X)) 24.81/8.97 top(ok(X)) -> top(active(X)) 24.81/8.97 24.81/8.97 S is empty. 24.81/8.97 Rewrite Strategy: FULL 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 24.81/8.97 The following defined symbols can occur below the 0th argument of top: proper, active 24.81/8.97 The following defined symbols can occur below the 0th argument of proper: proper, active 24.81/8.97 The following defined symbols can occur below the 0th argument of active: proper, active 24.81/8.97 24.81/8.97 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 24.81/8.97 active(first(0, X)) -> mark(nil) 24.81/8.97 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 24.81/8.97 active(from(X)) -> mark(cons(X, from(s(X)))) 24.81/8.97 active(first(X1, X2)) -> first(active(X1), X2) 24.81/8.97 active(first(X1, X2)) -> first(X1, active(X2)) 24.81/8.97 active(s(X)) -> s(active(X)) 24.81/8.97 active(cons(X1, X2)) -> cons(active(X1), X2) 24.81/8.97 active(from(X)) -> from(active(X)) 24.81/8.97 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 24.81/8.97 proper(s(X)) -> s(proper(X)) 24.81/8.97 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.81/8.97 proper(from(X)) -> from(proper(X)) 24.81/8.97 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (2) 24.81/8.97 Obligation: 24.81/8.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.81/8.97 24.81/8.97 24.81/8.97 The TRS R consists of the following rules: 24.81/8.97 24.81/8.97 first(mark(X1), X2) -> mark(first(X1, X2)) 24.81/8.97 first(X1, mark(X2)) -> mark(first(X1, X2)) 24.81/8.97 s(mark(X)) -> mark(s(X)) 24.81/8.97 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.81/8.97 from(mark(X)) -> mark(from(X)) 24.81/8.97 proper(0) -> ok(0) 24.81/8.97 proper(nil) -> ok(nil) 24.81/8.97 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 24.81/8.97 s(ok(X)) -> ok(s(X)) 24.81/8.97 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.81/8.97 from(ok(X)) -> ok(from(X)) 24.81/8.97 top(mark(X)) -> top(proper(X)) 24.81/8.97 top(ok(X)) -> top(active(X)) 24.81/8.97 24.81/8.97 S is empty. 24.81/8.97 Rewrite Strategy: FULL 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 24.81/8.97 transformed relative TRS to TRS 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (4) 24.81/8.97 Obligation: 24.81/8.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 24.81/8.97 24.81/8.97 24.81/8.97 The TRS R consists of the following rules: 24.81/8.97 24.81/8.97 first(mark(X1), X2) -> mark(first(X1, X2)) 24.81/8.97 first(X1, mark(X2)) -> mark(first(X1, X2)) 24.81/8.97 s(mark(X)) -> mark(s(X)) 24.81/8.97 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.81/8.97 from(mark(X)) -> mark(from(X)) 24.81/8.97 proper(0) -> ok(0) 24.81/8.97 proper(nil) -> ok(nil) 24.81/8.97 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 24.81/8.97 s(ok(X)) -> ok(s(X)) 24.81/8.97 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.81/8.97 from(ok(X)) -> ok(from(X)) 24.81/8.97 top(mark(X)) -> top(proper(X)) 24.81/8.97 top(ok(X)) -> top(active(X)) 24.81/8.97 24.81/8.97 S is empty. 24.81/8.97 Rewrite Strategy: FULL 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (5) CpxTrsMatchBoundsTAProof (FINISHED) 24.81/8.97 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. 24.81/8.97 24.81/8.97 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 24.81/8.97 final states : [1, 2, 3, 4, 5, 6] 24.81/8.97 transitions: 24.81/8.97 mark0(0) -> 0 24.81/8.97 00() -> 0 24.81/8.97 ok0(0) -> 0 24.81/8.97 nil0() -> 0 24.81/8.97 active0(0) -> 0 24.81/8.97 first0(0, 0) -> 1 24.81/8.97 s0(0) -> 2 24.81/8.97 cons0(0, 0) -> 3 24.81/8.97 from0(0) -> 4 24.81/8.97 proper0(0) -> 5 24.81/8.97 top0(0) -> 6 24.81/8.97 first1(0, 0) -> 7 24.81/8.97 mark1(7) -> 1 24.81/8.97 s1(0) -> 8 24.81/8.97 mark1(8) -> 2 24.81/8.97 cons1(0, 0) -> 9 24.81/8.97 mark1(9) -> 3 24.81/8.97 from1(0) -> 10 24.81/8.97 mark1(10) -> 4 24.81/8.97 01() -> 11 24.81/8.97 ok1(11) -> 5 24.81/8.97 nil1() -> 12 24.81/8.97 ok1(12) -> 5 24.81/8.97 first1(0, 0) -> 13 24.81/8.97 ok1(13) -> 1 24.81/8.97 s1(0) -> 14 24.81/8.97 ok1(14) -> 2 24.81/8.97 cons1(0, 0) -> 15 24.81/8.97 ok1(15) -> 3 24.81/8.97 from1(0) -> 16 24.81/8.97 ok1(16) -> 4 24.81/8.97 proper1(0) -> 17 24.81/8.97 top1(17) -> 6 24.81/8.97 active1(0) -> 18 24.81/8.97 top1(18) -> 6 24.81/8.97 mark1(7) -> 7 24.81/8.97 mark1(7) -> 13 24.81/8.97 mark1(8) -> 8 24.81/8.97 mark1(8) -> 14 24.81/8.97 mark1(9) -> 9 24.81/8.97 mark1(9) -> 15 24.81/8.97 mark1(10) -> 10 24.81/8.97 mark1(10) -> 16 24.81/8.97 ok1(11) -> 17 24.81/8.97 ok1(12) -> 17 24.81/8.97 ok1(13) -> 7 24.81/8.97 ok1(13) -> 13 24.81/8.97 ok1(14) -> 8 24.81/8.97 ok1(14) -> 14 24.81/8.97 ok1(15) -> 9 24.81/8.97 ok1(15) -> 15 24.81/8.97 ok1(16) -> 10 24.81/8.97 ok1(16) -> 16 24.81/8.97 active2(11) -> 19 24.81/8.97 top2(19) -> 6 24.81/8.97 active2(12) -> 19 24.81/8.97 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (6) 24.81/8.97 BOUNDS(1, n^1) 24.81/8.97 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 24.81/8.97 Transformed a relative TRS into a decreasing-loop problem. 24.81/8.97 ---------------------------------------- 24.81/8.97 24.81/8.97 (8) 24.81/8.97 Obligation: 24.81/8.97 Analyzing the following TRS for decreasing loops: 24.81/8.97 24.81/8.97 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.81/8.97 24.81/8.97 24.81/8.97 The TRS R consists of the following rules: 24.81/8.98 24.81/8.98 active(first(0, X)) -> mark(nil) 24.81/8.98 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 24.81/8.98 active(from(X)) -> mark(cons(X, from(s(X)))) 24.81/8.98 active(first(X1, X2)) -> first(active(X1), X2) 24.81/8.98 active(first(X1, X2)) -> first(X1, active(X2)) 24.81/8.98 active(s(X)) -> s(active(X)) 24.81/8.98 active(cons(X1, X2)) -> cons(active(X1), X2) 24.81/8.98 active(from(X)) -> from(active(X)) 24.81/8.98 first(mark(X1), X2) -> mark(first(X1, X2)) 24.81/8.98 first(X1, mark(X2)) -> mark(first(X1, X2)) 24.81/8.98 s(mark(X)) -> mark(s(X)) 24.81/8.98 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.81/8.98 from(mark(X)) -> mark(from(X)) 24.81/8.98 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 24.81/8.98 proper(0) -> ok(0) 24.81/8.98 proper(nil) -> ok(nil) 24.81/8.98 proper(s(X)) -> s(proper(X)) 24.81/8.98 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.81/8.98 proper(from(X)) -> from(proper(X)) 24.81/8.98 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 24.81/8.98 s(ok(X)) -> ok(s(X)) 24.81/8.98 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.81/8.98 from(ok(X)) -> ok(from(X)) 24.81/8.98 top(mark(X)) -> top(proper(X)) 24.81/8.98 top(ok(X)) -> top(active(X)) 24.81/8.98 24.81/8.98 S is empty. 24.81/8.98 Rewrite Strategy: FULL 24.81/8.98 ---------------------------------------- 24.81/8.98 24.81/8.98 (9) DecreasingLoopProof (LOWER BOUND(ID)) 24.81/8.98 The following loop(s) give(s) rise to the lower bound Omega(n^1): 24.81/8.98 24.81/8.98 The rewrite sequence 24.81/8.98 24.81/8.98 s(mark(X)) ->^+ mark(s(X)) 24.81/8.98 24.81/8.98 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 24.81/8.98 24.81/8.98 The pumping substitution is [X / mark(X)]. 24.81/8.98 24.81/8.98 The result substitution is [ ]. 24.81/8.98 24.81/8.98 24.81/8.98 24.81/8.98 24.81/8.98 ---------------------------------------- 24.81/8.98 24.81/8.98 (10) 24.81/8.98 Complex Obligation (BEST) 24.81/8.98 24.81/8.98 ---------------------------------------- 24.81/8.98 24.81/8.98 (11) 24.81/8.98 Obligation: 24.81/8.98 Proved the lower bound n^1 for the following obligation: 24.81/8.98 24.81/8.98 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.81/8.98 24.81/8.98 24.81/8.98 The TRS R consists of the following rules: 24.81/8.98 24.81/8.98 active(first(0, X)) -> mark(nil) 24.81/8.98 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 24.81/8.98 active(from(X)) -> mark(cons(X, from(s(X)))) 24.81/8.98 active(first(X1, X2)) -> first(active(X1), X2) 24.81/8.98 active(first(X1, X2)) -> first(X1, active(X2)) 24.81/8.98 active(s(X)) -> s(active(X)) 24.81/8.98 active(cons(X1, X2)) -> cons(active(X1), X2) 24.81/8.98 active(from(X)) -> from(active(X)) 24.81/8.98 first(mark(X1), X2) -> mark(first(X1, X2)) 24.81/8.98 first(X1, mark(X2)) -> mark(first(X1, X2)) 24.81/8.98 s(mark(X)) -> mark(s(X)) 24.81/8.98 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.81/8.98 from(mark(X)) -> mark(from(X)) 24.81/8.98 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 24.81/8.98 proper(0) -> ok(0) 24.81/8.98 proper(nil) -> ok(nil) 24.81/8.98 proper(s(X)) -> s(proper(X)) 24.81/8.98 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.81/8.98 proper(from(X)) -> from(proper(X)) 24.81/8.98 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 24.81/8.98 s(ok(X)) -> ok(s(X)) 24.81/8.98 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.81/8.98 from(ok(X)) -> ok(from(X)) 24.81/8.98 top(mark(X)) -> top(proper(X)) 24.81/8.98 top(ok(X)) -> top(active(X)) 24.81/8.98 24.81/8.98 S is empty. 24.81/8.98 Rewrite Strategy: FULL 24.81/8.98 ---------------------------------------- 24.81/8.98 24.81/8.98 (12) LowerBoundPropagationProof (FINISHED) 24.81/8.98 Propagated lower bound. 24.81/8.98 ---------------------------------------- 24.81/8.98 24.81/8.98 (13) 24.81/8.98 BOUNDS(n^1, INF) 24.81/8.98 24.81/8.98 ---------------------------------------- 24.81/8.98 24.81/8.98 (14) 24.81/8.98 Obligation: 24.81/8.98 Analyzing the following TRS for decreasing loops: 24.81/8.98 24.81/8.98 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 24.81/8.98 24.81/8.98 24.81/8.98 The TRS R consists of the following rules: 24.81/8.98 24.81/8.98 active(first(0, X)) -> mark(nil) 24.81/8.98 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 24.81/8.98 active(from(X)) -> mark(cons(X, from(s(X)))) 24.81/8.98 active(first(X1, X2)) -> first(active(X1), X2) 24.81/8.98 active(first(X1, X2)) -> first(X1, active(X2)) 24.81/8.98 active(s(X)) -> s(active(X)) 24.81/8.98 active(cons(X1, X2)) -> cons(active(X1), X2) 24.81/8.98 active(from(X)) -> from(active(X)) 24.81/8.98 first(mark(X1), X2) -> mark(first(X1, X2)) 24.81/8.98 first(X1, mark(X2)) -> mark(first(X1, X2)) 24.81/8.98 s(mark(X)) -> mark(s(X)) 24.81/8.98 cons(mark(X1), X2) -> mark(cons(X1, X2)) 24.81/8.98 from(mark(X)) -> mark(from(X)) 24.81/8.98 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 24.81/8.98 proper(0) -> ok(0) 24.81/8.98 proper(nil) -> ok(nil) 24.81/8.98 proper(s(X)) -> s(proper(X)) 24.81/8.98 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 24.81/8.98 proper(from(X)) -> from(proper(X)) 24.81/8.98 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 24.81/8.98 s(ok(X)) -> ok(s(X)) 24.81/8.98 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 24.81/8.98 from(ok(X)) -> ok(from(X)) 24.81/8.98 top(mark(X)) -> top(proper(X)) 24.81/8.98 top(ok(X)) -> top(active(X)) 24.81/8.98 24.81/8.98 S is empty. 24.81/8.98 Rewrite Strategy: FULL 25.02/9.03 EOF