38.35/11.79 WORST_CASE(Omega(n^1), O(n^1)) 38.35/11.80 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 38.35/11.80 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 38.35/11.80 38.35/11.80 38.35/11.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 38.35/11.80 38.35/11.80 (0) CpxTRS 38.35/11.80 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 13 ms] 38.35/11.80 (2) CpxTRS 38.35/11.80 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 38.35/11.80 (4) CpxTRS 38.35/11.80 (5) CpxTrsMatchBoundsTAProof [FINISHED, 102 ms] 38.35/11.80 (6) BOUNDS(1, n^1) 38.35/11.80 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 38.35/11.80 (8) TRS for Loop Detection 38.35/11.80 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 38.35/11.80 (10) BEST 38.35/11.80 (11) proven lower bound 38.35/11.80 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 38.35/11.80 (13) BOUNDS(n^1, INF) 38.35/11.80 (14) TRS for Loop Detection 38.35/11.80 38.35/11.80 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (0) 38.35/11.80 Obligation: 38.35/11.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 38.35/11.80 38.35/11.80 38.35/11.80 The TRS R consists of the following rules: 38.35/11.80 38.35/11.80 active(from(X)) -> mark(cons(X, from(s(X)))) 38.35/11.80 active(sel(0, cons(X, XS))) -> mark(X) 38.35/11.80 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 38.35/11.80 active(minus(X, 0)) -> mark(0) 38.35/11.80 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 38.35/11.80 active(quot(0, s(Y))) -> mark(0) 38.35/11.80 active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y)))) 38.35/11.80 active(zWquot(XS, nil)) -> mark(nil) 38.35/11.80 active(zWquot(nil, XS)) -> mark(nil) 38.35/11.80 active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS))) 38.35/11.80 active(from(X)) -> from(active(X)) 38.35/11.80 active(cons(X1, X2)) -> cons(active(X1), X2) 38.35/11.80 active(s(X)) -> s(active(X)) 38.35/11.80 active(sel(X1, X2)) -> sel(active(X1), X2) 38.35/11.80 active(sel(X1, X2)) -> sel(X1, active(X2)) 38.35/11.80 active(minus(X1, X2)) -> minus(active(X1), X2) 38.35/11.80 active(minus(X1, X2)) -> minus(X1, active(X2)) 38.35/11.80 active(quot(X1, X2)) -> quot(active(X1), X2) 38.35/11.80 active(quot(X1, X2)) -> quot(X1, active(X2)) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(active(X1), X2) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(X1, active(X2)) 38.35/11.80 from(mark(X)) -> mark(from(X)) 38.35/11.80 cons(mark(X1), X2) -> mark(cons(X1, X2)) 38.35/11.80 s(mark(X)) -> mark(s(X)) 38.35/11.80 sel(mark(X1), X2) -> mark(sel(X1, X2)) 38.35/11.80 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 38.35/11.80 minus(mark(X1), X2) -> mark(minus(X1, X2)) 38.35/11.80 minus(X1, mark(X2)) -> mark(minus(X1, X2)) 38.35/11.80 quot(mark(X1), X2) -> mark(quot(X1, X2)) 38.35/11.80 quot(X1, mark(X2)) -> mark(quot(X1, X2)) 38.35/11.80 zWquot(mark(X1), X2) -> mark(zWquot(X1, X2)) 38.35/11.80 zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2)) 38.35/11.80 proper(from(X)) -> from(proper(X)) 38.35/11.80 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 38.35/11.80 proper(s(X)) -> s(proper(X)) 38.35/11.80 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 38.35/11.80 proper(0) -> ok(0) 38.35/11.80 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 38.35/11.80 proper(quot(X1, X2)) -> quot(proper(X1), proper(X2)) 38.35/11.80 proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2)) 38.35/11.80 proper(nil) -> ok(nil) 38.35/11.80 from(ok(X)) -> ok(from(X)) 38.35/11.80 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 38.35/11.80 s(ok(X)) -> ok(s(X)) 38.35/11.80 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 38.35/11.80 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 38.35/11.80 quot(ok(X1), ok(X2)) -> ok(quot(X1, X2)) 38.35/11.80 zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2)) 38.35/11.80 top(mark(X)) -> top(proper(X)) 38.35/11.80 top(ok(X)) -> top(active(X)) 38.35/11.80 38.35/11.80 S is empty. 38.35/11.80 Rewrite Strategy: FULL 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 38.35/11.80 The following defined symbols can occur below the 0th argument of top: proper, active 38.35/11.80 The following defined symbols can occur below the 0th argument of proper: proper, active 38.35/11.80 The following defined symbols can occur below the 0th argument of active: proper, active 38.35/11.80 38.35/11.80 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 38.35/11.80 active(from(X)) -> mark(cons(X, from(s(X)))) 38.35/11.80 active(sel(0, cons(X, XS))) -> mark(X) 38.35/11.80 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 38.35/11.80 active(minus(X, 0)) -> mark(0) 38.35/11.80 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 38.35/11.80 active(quot(0, s(Y))) -> mark(0) 38.35/11.80 active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y)))) 38.35/11.80 active(zWquot(XS, nil)) -> mark(nil) 38.35/11.80 active(zWquot(nil, XS)) -> mark(nil) 38.35/11.80 active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS))) 38.35/11.80 active(from(X)) -> from(active(X)) 38.35/11.80 active(cons(X1, X2)) -> cons(active(X1), X2) 38.35/11.80 active(s(X)) -> s(active(X)) 38.35/11.80 active(sel(X1, X2)) -> sel(active(X1), X2) 38.35/11.80 active(sel(X1, X2)) -> sel(X1, active(X2)) 38.35/11.80 active(minus(X1, X2)) -> minus(active(X1), X2) 38.35/11.80 active(minus(X1, X2)) -> minus(X1, active(X2)) 38.35/11.80 active(quot(X1, X2)) -> quot(active(X1), X2) 38.35/11.80 active(quot(X1, X2)) -> quot(X1, active(X2)) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(active(X1), X2) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(X1, active(X2)) 38.35/11.80 proper(from(X)) -> from(proper(X)) 38.35/11.80 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 38.35/11.80 proper(s(X)) -> s(proper(X)) 38.35/11.80 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 38.35/11.80 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 38.35/11.80 proper(quot(X1, X2)) -> quot(proper(X1), proper(X2)) 38.35/11.80 proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2)) 38.35/11.80 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (2) 38.35/11.80 Obligation: 38.35/11.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 38.35/11.80 38.35/11.80 38.35/11.80 The TRS R consists of the following rules: 38.35/11.80 38.35/11.80 from(mark(X)) -> mark(from(X)) 38.35/11.80 cons(mark(X1), X2) -> mark(cons(X1, X2)) 38.35/11.80 s(mark(X)) -> mark(s(X)) 38.35/11.80 sel(mark(X1), X2) -> mark(sel(X1, X2)) 38.35/11.80 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 38.35/11.80 minus(mark(X1), X2) -> mark(minus(X1, X2)) 38.35/11.80 minus(X1, mark(X2)) -> mark(minus(X1, X2)) 38.35/11.80 quot(mark(X1), X2) -> mark(quot(X1, X2)) 38.35/11.80 quot(X1, mark(X2)) -> mark(quot(X1, X2)) 38.35/11.80 zWquot(mark(X1), X2) -> mark(zWquot(X1, X2)) 38.35/11.80 zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2)) 38.35/11.80 proper(0) -> ok(0) 38.35/11.80 proper(nil) -> ok(nil) 38.35/11.80 from(ok(X)) -> ok(from(X)) 38.35/11.80 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 38.35/11.80 s(ok(X)) -> ok(s(X)) 38.35/11.80 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 38.35/11.80 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 38.35/11.80 quot(ok(X1), ok(X2)) -> ok(quot(X1, X2)) 38.35/11.80 zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2)) 38.35/11.80 top(mark(X)) -> top(proper(X)) 38.35/11.80 top(ok(X)) -> top(active(X)) 38.35/11.80 38.35/11.80 S is empty. 38.35/11.80 Rewrite Strategy: FULL 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 38.35/11.80 transformed relative TRS to TRS 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (4) 38.35/11.80 Obligation: 38.35/11.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 38.35/11.80 38.35/11.80 38.35/11.80 The TRS R consists of the following rules: 38.35/11.80 38.35/11.80 from(mark(X)) -> mark(from(X)) 38.35/11.80 cons(mark(X1), X2) -> mark(cons(X1, X2)) 38.35/11.80 s(mark(X)) -> mark(s(X)) 38.35/11.80 sel(mark(X1), X2) -> mark(sel(X1, X2)) 38.35/11.80 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 38.35/11.80 minus(mark(X1), X2) -> mark(minus(X1, X2)) 38.35/11.80 minus(X1, mark(X2)) -> mark(minus(X1, X2)) 38.35/11.80 quot(mark(X1), X2) -> mark(quot(X1, X2)) 38.35/11.80 quot(X1, mark(X2)) -> mark(quot(X1, X2)) 38.35/11.80 zWquot(mark(X1), X2) -> mark(zWquot(X1, X2)) 38.35/11.80 zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2)) 38.35/11.80 proper(0) -> ok(0) 38.35/11.80 proper(nil) -> ok(nil) 38.35/11.80 from(ok(X)) -> ok(from(X)) 38.35/11.80 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 38.35/11.80 s(ok(X)) -> ok(s(X)) 38.35/11.80 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 38.35/11.80 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 38.35/11.80 quot(ok(X1), ok(X2)) -> ok(quot(X1, X2)) 38.35/11.80 zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2)) 38.35/11.80 top(mark(X)) -> top(proper(X)) 38.35/11.80 top(ok(X)) -> top(active(X)) 38.35/11.80 38.35/11.80 S is empty. 38.35/11.80 Rewrite Strategy: FULL 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (5) CpxTrsMatchBoundsTAProof (FINISHED) 38.35/11.80 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. 38.35/11.80 38.35/11.80 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 38.35/11.80 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9] 38.35/11.80 transitions: 38.35/11.80 mark0(0) -> 0 38.35/11.80 00() -> 0 38.35/11.80 ok0(0) -> 0 38.35/11.80 nil0() -> 0 38.35/11.80 active0(0) -> 0 38.35/11.80 from0(0) -> 1 38.35/11.80 cons0(0, 0) -> 2 38.35/11.80 s0(0) -> 3 38.35/11.80 sel0(0, 0) -> 4 38.35/11.80 minus0(0, 0) -> 5 38.35/11.80 quot0(0, 0) -> 6 38.35/11.80 zWquot0(0, 0) -> 7 38.35/11.80 proper0(0) -> 8 38.35/11.80 top0(0) -> 9 38.35/11.80 from1(0) -> 10 38.35/11.80 mark1(10) -> 1 38.35/11.80 cons1(0, 0) -> 11 38.35/11.80 mark1(11) -> 2 38.35/11.80 s1(0) -> 12 38.35/11.80 mark1(12) -> 3 38.35/11.80 sel1(0, 0) -> 13 38.35/11.80 mark1(13) -> 4 38.35/11.80 minus1(0, 0) -> 14 38.35/11.80 mark1(14) -> 5 38.35/11.80 quot1(0, 0) -> 15 38.35/11.80 mark1(15) -> 6 38.35/11.80 zWquot1(0, 0) -> 16 38.35/11.80 mark1(16) -> 7 38.35/11.80 01() -> 17 38.35/11.80 ok1(17) -> 8 38.35/11.80 nil1() -> 18 38.35/11.80 ok1(18) -> 8 38.35/11.80 from1(0) -> 19 38.35/11.80 ok1(19) -> 1 38.35/11.80 cons1(0, 0) -> 20 38.35/11.80 ok1(20) -> 2 38.35/11.80 s1(0) -> 21 38.35/11.80 ok1(21) -> 3 38.35/11.80 sel1(0, 0) -> 22 38.35/11.80 ok1(22) -> 4 38.35/11.80 minus1(0, 0) -> 23 38.35/11.80 ok1(23) -> 5 38.35/11.80 quot1(0, 0) -> 24 38.35/11.80 ok1(24) -> 6 38.35/11.80 zWquot1(0, 0) -> 25 38.35/11.80 ok1(25) -> 7 38.35/11.80 proper1(0) -> 26 38.35/11.80 top1(26) -> 9 38.35/11.80 active1(0) -> 27 38.35/11.80 top1(27) -> 9 38.35/11.80 mark1(10) -> 10 38.35/11.80 mark1(10) -> 19 38.35/11.80 mark1(11) -> 11 38.35/11.80 mark1(11) -> 20 38.35/11.80 mark1(12) -> 12 38.35/11.80 mark1(12) -> 21 38.35/11.80 mark1(13) -> 13 38.35/11.80 mark1(13) -> 22 38.35/11.80 mark1(14) -> 14 38.35/11.80 mark1(14) -> 23 38.35/11.80 mark1(15) -> 15 38.35/11.80 mark1(15) -> 24 38.35/11.80 mark1(16) -> 16 38.35/11.80 mark1(16) -> 25 38.35/11.80 ok1(17) -> 26 38.35/11.80 ok1(18) -> 26 38.35/11.80 ok1(19) -> 10 38.35/11.80 ok1(19) -> 19 38.35/11.80 ok1(20) -> 11 38.35/11.80 ok1(20) -> 20 38.35/11.80 ok1(21) -> 12 38.35/11.80 ok1(21) -> 21 38.35/11.80 ok1(22) -> 13 38.35/11.80 ok1(22) -> 22 38.35/11.80 ok1(23) -> 14 38.35/11.80 ok1(23) -> 23 38.35/11.80 ok1(24) -> 15 38.35/11.80 ok1(24) -> 24 38.35/11.80 ok1(25) -> 16 38.35/11.80 ok1(25) -> 25 38.35/11.80 active2(17) -> 28 38.35/11.80 top2(28) -> 9 38.35/11.80 active2(18) -> 28 38.35/11.80 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (6) 38.35/11.80 BOUNDS(1, n^1) 38.35/11.80 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 38.35/11.80 Transformed a relative TRS into a decreasing-loop problem. 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (8) 38.35/11.80 Obligation: 38.35/11.80 Analyzing the following TRS for decreasing loops: 38.35/11.80 38.35/11.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 38.35/11.80 38.35/11.80 38.35/11.80 The TRS R consists of the following rules: 38.35/11.80 38.35/11.80 active(from(X)) -> mark(cons(X, from(s(X)))) 38.35/11.80 active(sel(0, cons(X, XS))) -> mark(X) 38.35/11.80 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 38.35/11.80 active(minus(X, 0)) -> mark(0) 38.35/11.80 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 38.35/11.80 active(quot(0, s(Y))) -> mark(0) 38.35/11.80 active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y)))) 38.35/11.80 active(zWquot(XS, nil)) -> mark(nil) 38.35/11.80 active(zWquot(nil, XS)) -> mark(nil) 38.35/11.80 active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS))) 38.35/11.80 active(from(X)) -> from(active(X)) 38.35/11.80 active(cons(X1, X2)) -> cons(active(X1), X2) 38.35/11.80 active(s(X)) -> s(active(X)) 38.35/11.80 active(sel(X1, X2)) -> sel(active(X1), X2) 38.35/11.80 active(sel(X1, X2)) -> sel(X1, active(X2)) 38.35/11.80 active(minus(X1, X2)) -> minus(active(X1), X2) 38.35/11.80 active(minus(X1, X2)) -> minus(X1, active(X2)) 38.35/11.80 active(quot(X1, X2)) -> quot(active(X1), X2) 38.35/11.80 active(quot(X1, X2)) -> quot(X1, active(X2)) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(active(X1), X2) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(X1, active(X2)) 38.35/11.80 from(mark(X)) -> mark(from(X)) 38.35/11.80 cons(mark(X1), X2) -> mark(cons(X1, X2)) 38.35/11.80 s(mark(X)) -> mark(s(X)) 38.35/11.80 sel(mark(X1), X2) -> mark(sel(X1, X2)) 38.35/11.80 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 38.35/11.80 minus(mark(X1), X2) -> mark(minus(X1, X2)) 38.35/11.80 minus(X1, mark(X2)) -> mark(minus(X1, X2)) 38.35/11.80 quot(mark(X1), X2) -> mark(quot(X1, X2)) 38.35/11.80 quot(X1, mark(X2)) -> mark(quot(X1, X2)) 38.35/11.80 zWquot(mark(X1), X2) -> mark(zWquot(X1, X2)) 38.35/11.80 zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2)) 38.35/11.80 proper(from(X)) -> from(proper(X)) 38.35/11.80 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 38.35/11.80 proper(s(X)) -> s(proper(X)) 38.35/11.80 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 38.35/11.80 proper(0) -> ok(0) 38.35/11.80 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 38.35/11.80 proper(quot(X1, X2)) -> quot(proper(X1), proper(X2)) 38.35/11.80 proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2)) 38.35/11.80 proper(nil) -> ok(nil) 38.35/11.80 from(ok(X)) -> ok(from(X)) 38.35/11.80 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 38.35/11.80 s(ok(X)) -> ok(s(X)) 38.35/11.80 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 38.35/11.80 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 38.35/11.80 quot(ok(X1), ok(X2)) -> ok(quot(X1, X2)) 38.35/11.80 zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2)) 38.35/11.80 top(mark(X)) -> top(proper(X)) 38.35/11.80 top(ok(X)) -> top(active(X)) 38.35/11.80 38.35/11.80 S is empty. 38.35/11.80 Rewrite Strategy: FULL 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (9) DecreasingLoopProof (LOWER BOUND(ID)) 38.35/11.80 The following loop(s) give(s) rise to the lower bound Omega(n^1): 38.35/11.80 38.35/11.80 The rewrite sequence 38.35/11.80 38.35/11.80 quot(ok(X1), ok(X2)) ->^+ ok(quot(X1, X2)) 38.35/11.80 38.35/11.80 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 38.35/11.80 38.35/11.80 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 38.35/11.80 38.35/11.80 The result substitution is [ ]. 38.35/11.80 38.35/11.80 38.35/11.80 38.35/11.80 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (10) 38.35/11.80 Complex Obligation (BEST) 38.35/11.80 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (11) 38.35/11.80 Obligation: 38.35/11.80 Proved the lower bound n^1 for the following obligation: 38.35/11.80 38.35/11.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 38.35/11.80 38.35/11.80 38.35/11.80 The TRS R consists of the following rules: 38.35/11.80 38.35/11.80 active(from(X)) -> mark(cons(X, from(s(X)))) 38.35/11.80 active(sel(0, cons(X, XS))) -> mark(X) 38.35/11.80 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 38.35/11.80 active(minus(X, 0)) -> mark(0) 38.35/11.80 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 38.35/11.80 active(quot(0, s(Y))) -> mark(0) 38.35/11.80 active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y)))) 38.35/11.80 active(zWquot(XS, nil)) -> mark(nil) 38.35/11.80 active(zWquot(nil, XS)) -> mark(nil) 38.35/11.80 active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS))) 38.35/11.80 active(from(X)) -> from(active(X)) 38.35/11.80 active(cons(X1, X2)) -> cons(active(X1), X2) 38.35/11.80 active(s(X)) -> s(active(X)) 38.35/11.80 active(sel(X1, X2)) -> sel(active(X1), X2) 38.35/11.80 active(sel(X1, X2)) -> sel(X1, active(X2)) 38.35/11.80 active(minus(X1, X2)) -> minus(active(X1), X2) 38.35/11.80 active(minus(X1, X2)) -> minus(X1, active(X2)) 38.35/11.80 active(quot(X1, X2)) -> quot(active(X1), X2) 38.35/11.80 active(quot(X1, X2)) -> quot(X1, active(X2)) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(active(X1), X2) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(X1, active(X2)) 38.35/11.80 from(mark(X)) -> mark(from(X)) 38.35/11.80 cons(mark(X1), X2) -> mark(cons(X1, X2)) 38.35/11.80 s(mark(X)) -> mark(s(X)) 38.35/11.80 sel(mark(X1), X2) -> mark(sel(X1, X2)) 38.35/11.80 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 38.35/11.80 minus(mark(X1), X2) -> mark(minus(X1, X2)) 38.35/11.80 minus(X1, mark(X2)) -> mark(minus(X1, X2)) 38.35/11.80 quot(mark(X1), X2) -> mark(quot(X1, X2)) 38.35/11.80 quot(X1, mark(X2)) -> mark(quot(X1, X2)) 38.35/11.80 zWquot(mark(X1), X2) -> mark(zWquot(X1, X2)) 38.35/11.80 zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2)) 38.35/11.80 proper(from(X)) -> from(proper(X)) 38.35/11.80 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 38.35/11.80 proper(s(X)) -> s(proper(X)) 38.35/11.80 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 38.35/11.80 proper(0) -> ok(0) 38.35/11.80 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 38.35/11.80 proper(quot(X1, X2)) -> quot(proper(X1), proper(X2)) 38.35/11.80 proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2)) 38.35/11.80 proper(nil) -> ok(nil) 38.35/11.80 from(ok(X)) -> ok(from(X)) 38.35/11.80 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 38.35/11.80 s(ok(X)) -> ok(s(X)) 38.35/11.80 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 38.35/11.80 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 38.35/11.80 quot(ok(X1), ok(X2)) -> ok(quot(X1, X2)) 38.35/11.80 zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2)) 38.35/11.80 top(mark(X)) -> top(proper(X)) 38.35/11.80 top(ok(X)) -> top(active(X)) 38.35/11.80 38.35/11.80 S is empty. 38.35/11.80 Rewrite Strategy: FULL 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (12) LowerBoundPropagationProof (FINISHED) 38.35/11.80 Propagated lower bound. 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (13) 38.35/11.80 BOUNDS(n^1, INF) 38.35/11.80 38.35/11.80 ---------------------------------------- 38.35/11.80 38.35/11.80 (14) 38.35/11.80 Obligation: 38.35/11.80 Analyzing the following TRS for decreasing loops: 38.35/11.80 38.35/11.80 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 38.35/11.80 38.35/11.80 38.35/11.80 The TRS R consists of the following rules: 38.35/11.80 38.35/11.80 active(from(X)) -> mark(cons(X, from(s(X)))) 38.35/11.80 active(sel(0, cons(X, XS))) -> mark(X) 38.35/11.80 active(sel(s(N), cons(X, XS))) -> mark(sel(N, XS)) 38.35/11.80 active(minus(X, 0)) -> mark(0) 38.35/11.80 active(minus(s(X), s(Y))) -> mark(minus(X, Y)) 38.35/11.80 active(quot(0, s(Y))) -> mark(0) 38.35/11.80 active(quot(s(X), s(Y))) -> mark(s(quot(minus(X, Y), s(Y)))) 38.35/11.80 active(zWquot(XS, nil)) -> mark(nil) 38.35/11.80 active(zWquot(nil, XS)) -> mark(nil) 38.35/11.80 active(zWquot(cons(X, XS), cons(Y, YS))) -> mark(cons(quot(X, Y), zWquot(XS, YS))) 38.35/11.80 active(from(X)) -> from(active(X)) 38.35/11.80 active(cons(X1, X2)) -> cons(active(X1), X2) 38.35/11.80 active(s(X)) -> s(active(X)) 38.35/11.80 active(sel(X1, X2)) -> sel(active(X1), X2) 38.35/11.80 active(sel(X1, X2)) -> sel(X1, active(X2)) 38.35/11.80 active(minus(X1, X2)) -> minus(active(X1), X2) 38.35/11.80 active(minus(X1, X2)) -> minus(X1, active(X2)) 38.35/11.80 active(quot(X1, X2)) -> quot(active(X1), X2) 38.35/11.80 active(quot(X1, X2)) -> quot(X1, active(X2)) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(active(X1), X2) 38.35/11.80 active(zWquot(X1, X2)) -> zWquot(X1, active(X2)) 38.35/11.80 from(mark(X)) -> mark(from(X)) 38.35/11.80 cons(mark(X1), X2) -> mark(cons(X1, X2)) 38.35/11.80 s(mark(X)) -> mark(s(X)) 38.35/11.80 sel(mark(X1), X2) -> mark(sel(X1, X2)) 38.35/11.80 sel(X1, mark(X2)) -> mark(sel(X1, X2)) 38.35/11.80 minus(mark(X1), X2) -> mark(minus(X1, X2)) 38.35/11.80 minus(X1, mark(X2)) -> mark(minus(X1, X2)) 38.35/11.80 quot(mark(X1), X2) -> mark(quot(X1, X2)) 38.35/11.80 quot(X1, mark(X2)) -> mark(quot(X1, X2)) 38.35/11.80 zWquot(mark(X1), X2) -> mark(zWquot(X1, X2)) 38.35/11.80 zWquot(X1, mark(X2)) -> mark(zWquot(X1, X2)) 38.35/11.80 proper(from(X)) -> from(proper(X)) 38.35/11.80 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 38.35/11.80 proper(s(X)) -> s(proper(X)) 38.35/11.80 proper(sel(X1, X2)) -> sel(proper(X1), proper(X2)) 38.35/11.80 proper(0) -> ok(0) 38.35/11.80 proper(minus(X1, X2)) -> minus(proper(X1), proper(X2)) 38.35/11.80 proper(quot(X1, X2)) -> quot(proper(X1), proper(X2)) 38.35/11.80 proper(zWquot(X1, X2)) -> zWquot(proper(X1), proper(X2)) 38.35/11.80 proper(nil) -> ok(nil) 38.35/11.80 from(ok(X)) -> ok(from(X)) 38.35/11.80 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 38.35/11.80 s(ok(X)) -> ok(s(X)) 38.35/11.80 sel(ok(X1), ok(X2)) -> ok(sel(X1, X2)) 38.35/11.80 minus(ok(X1), ok(X2)) -> ok(minus(X1, X2)) 38.35/11.80 quot(ok(X1), ok(X2)) -> ok(quot(X1, X2)) 38.35/11.80 zWquot(ok(X1), ok(X2)) -> ok(zWquot(X1, X2)) 38.35/11.80 top(mark(X)) -> top(proper(X)) 38.35/11.80 top(ok(X)) -> top(active(X)) 38.35/11.80 38.35/11.80 S is empty. 38.35/11.80 Rewrite Strategy: FULL 38.43/11.84 EOF