41.72/12.35 WORST_CASE(Omega(n^1), O(n^1)) 41.72/12.37 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 41.72/12.37 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 41.72/12.37 41.72/12.37 41.72/12.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.72/12.37 41.72/12.37 (0) CpxTRS 41.72/12.37 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 41.72/12.37 (2) CpxTRS 41.72/12.37 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 41.72/12.37 (4) CpxTRS 41.72/12.37 (5) CpxTrsMatchBoundsTAProof [FINISHED, 268 ms] 41.72/12.37 (6) BOUNDS(1, n^1) 41.72/12.37 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 41.72/12.37 (8) TRS for Loop Detection 41.72/12.37 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 41.72/12.37 (10) BEST 41.72/12.37 (11) proven lower bound 41.72/12.37 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 41.72/12.37 (13) BOUNDS(n^1, INF) 41.72/12.37 (14) TRS for Loop Detection 41.72/12.37 41.72/12.37 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (0) 41.72/12.37 Obligation: 41.72/12.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.72/12.37 41.72/12.37 41.72/12.37 The TRS R consists of the following rules: 41.72/12.37 41.72/12.37 active(from(X)) -> mark(cons(X, from(s(X)))) 41.72/12.37 active(2ndspos(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) 41.72/12.37 active(2ndsneg(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) 41.72/12.37 active(pi(X)) -> mark(2ndspos(X, from(0))) 41.72/12.37 active(plus(0, Y)) -> mark(Y) 41.72/12.37 active(plus(s(X), Y)) -> mark(s(plus(X, Y))) 41.72/12.37 active(times(0, Y)) -> mark(0) 41.72/12.37 active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) 41.72/12.37 active(square(X)) -> mark(times(X, X)) 41.72/12.37 active(s(X)) -> s(active(X)) 41.72/12.37 active(posrecip(X)) -> posrecip(active(X)) 41.72/12.37 active(negrecip(X)) -> negrecip(active(X)) 41.72/12.37 active(cons(X1, X2)) -> cons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(X1, active(X2)) 41.72/12.37 active(from(X)) -> from(active(X)) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) 41.72/12.37 active(pi(X)) -> pi(active(X)) 41.72/12.37 active(plus(X1, X2)) -> plus(active(X1), X2) 41.72/12.37 active(plus(X1, X2)) -> plus(X1, active(X2)) 41.72/12.37 active(times(X1, X2)) -> times(active(X1), X2) 41.72/12.37 active(times(X1, X2)) -> times(X1, active(X2)) 41.72/12.37 active(square(X)) -> square(active(X)) 41.72/12.37 s(mark(X)) -> mark(s(X)) 41.72/12.37 posrecip(mark(X)) -> mark(posrecip(X)) 41.72/12.37 negrecip(mark(X)) -> mark(negrecip(X)) 41.72/12.37 cons(mark(X1), X2) -> mark(cons(X1, X2)) 41.72/12.37 rcons(mark(X1), X2) -> mark(rcons(X1, X2)) 41.72/12.37 rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) 41.72/12.37 from(mark(X)) -> mark(from(X)) 41.72/12.37 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 41.72/12.37 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) 41.72/12.37 pi(mark(X)) -> mark(pi(X)) 41.72/12.37 plus(mark(X1), X2) -> mark(plus(X1, X2)) 41.72/12.37 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 41.72/12.37 times(mark(X1), X2) -> mark(times(X1, X2)) 41.72/12.37 times(X1, mark(X2)) -> mark(times(X1, X2)) 41.72/12.37 square(mark(X)) -> mark(square(X)) 41.72/12.37 proper(0) -> ok(0) 41.72/12.37 proper(s(X)) -> s(proper(X)) 41.72/12.37 proper(posrecip(X)) -> posrecip(proper(X)) 41.72/12.37 proper(negrecip(X)) -> negrecip(proper(X)) 41.72/12.37 proper(nil) -> ok(nil) 41.72/12.37 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 41.72/12.37 proper(rnil) -> ok(rnil) 41.72/12.37 proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) 41.72/12.37 proper(from(X)) -> from(proper(X)) 41.72/12.37 proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) 41.72/12.37 proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) 41.72/12.37 proper(pi(X)) -> pi(proper(X)) 41.72/12.37 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 41.72/12.37 proper(times(X1, X2)) -> times(proper(X1), proper(X2)) 41.72/12.37 proper(square(X)) -> square(proper(X)) 41.72/12.37 s(ok(X)) -> ok(s(X)) 41.72/12.37 posrecip(ok(X)) -> ok(posrecip(X)) 41.72/12.37 negrecip(ok(X)) -> ok(negrecip(X)) 41.72/12.37 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 41.72/12.37 rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) 41.72/12.37 from(ok(X)) -> ok(from(X)) 41.72/12.37 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) 41.72/12.37 pi(ok(X)) -> ok(pi(X)) 41.72/12.37 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 41.72/12.37 times(ok(X1), ok(X2)) -> ok(times(X1, X2)) 41.72/12.37 square(ok(X)) -> ok(square(X)) 41.72/12.37 top(mark(X)) -> top(proper(X)) 41.72/12.37 top(ok(X)) -> top(active(X)) 41.72/12.37 41.72/12.37 S is empty. 41.72/12.37 Rewrite Strategy: FULL 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 41.72/12.37 The following defined symbols can occur below the 0th argument of top: proper, active 41.72/12.37 The following defined symbols can occur below the 0th argument of proper: proper, active 41.72/12.37 The following defined symbols can occur below the 0th argument of active: proper, active 41.72/12.37 41.72/12.37 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 41.72/12.37 active(from(X)) -> mark(cons(X, from(s(X)))) 41.72/12.37 active(2ndspos(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) 41.72/12.37 active(2ndsneg(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) 41.72/12.37 active(pi(X)) -> mark(2ndspos(X, from(0))) 41.72/12.37 active(plus(0, Y)) -> mark(Y) 41.72/12.37 active(plus(s(X), Y)) -> mark(s(plus(X, Y))) 41.72/12.37 active(times(0, Y)) -> mark(0) 41.72/12.37 active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) 41.72/12.37 active(square(X)) -> mark(times(X, X)) 41.72/12.37 active(s(X)) -> s(active(X)) 41.72/12.37 active(posrecip(X)) -> posrecip(active(X)) 41.72/12.37 active(negrecip(X)) -> negrecip(active(X)) 41.72/12.37 active(cons(X1, X2)) -> cons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(X1, active(X2)) 41.72/12.37 active(from(X)) -> from(active(X)) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) 41.72/12.37 active(pi(X)) -> pi(active(X)) 41.72/12.37 active(plus(X1, X2)) -> plus(active(X1), X2) 41.72/12.37 active(plus(X1, X2)) -> plus(X1, active(X2)) 41.72/12.37 active(times(X1, X2)) -> times(active(X1), X2) 41.72/12.37 active(times(X1, X2)) -> times(X1, active(X2)) 41.72/12.37 active(square(X)) -> square(active(X)) 41.72/12.37 proper(s(X)) -> s(proper(X)) 41.72/12.37 proper(posrecip(X)) -> posrecip(proper(X)) 41.72/12.37 proper(negrecip(X)) -> negrecip(proper(X)) 41.72/12.37 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 41.72/12.37 proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) 41.72/12.37 proper(from(X)) -> from(proper(X)) 41.72/12.37 proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) 41.72/12.37 proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) 41.72/12.37 proper(pi(X)) -> pi(proper(X)) 41.72/12.37 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 41.72/12.37 proper(times(X1, X2)) -> times(proper(X1), proper(X2)) 41.72/12.37 proper(square(X)) -> square(proper(X)) 41.72/12.37 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (2) 41.72/12.37 Obligation: 41.72/12.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 41.72/12.37 41.72/12.37 41.72/12.37 The TRS R consists of the following rules: 41.72/12.37 41.72/12.37 s(mark(X)) -> mark(s(X)) 41.72/12.37 posrecip(mark(X)) -> mark(posrecip(X)) 41.72/12.37 negrecip(mark(X)) -> mark(negrecip(X)) 41.72/12.37 cons(mark(X1), X2) -> mark(cons(X1, X2)) 41.72/12.37 rcons(mark(X1), X2) -> mark(rcons(X1, X2)) 41.72/12.37 rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) 41.72/12.37 from(mark(X)) -> mark(from(X)) 41.72/12.37 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 41.72/12.37 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) 41.72/12.37 pi(mark(X)) -> mark(pi(X)) 41.72/12.37 plus(mark(X1), X2) -> mark(plus(X1, X2)) 41.72/12.37 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 41.72/12.37 times(mark(X1), X2) -> mark(times(X1, X2)) 41.72/12.37 times(X1, mark(X2)) -> mark(times(X1, X2)) 41.72/12.37 square(mark(X)) -> mark(square(X)) 41.72/12.37 proper(0) -> ok(0) 41.72/12.37 proper(nil) -> ok(nil) 41.72/12.37 proper(rnil) -> ok(rnil) 41.72/12.37 s(ok(X)) -> ok(s(X)) 41.72/12.37 posrecip(ok(X)) -> ok(posrecip(X)) 41.72/12.37 negrecip(ok(X)) -> ok(negrecip(X)) 41.72/12.37 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 41.72/12.37 rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) 41.72/12.37 from(ok(X)) -> ok(from(X)) 41.72/12.37 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) 41.72/12.37 pi(ok(X)) -> ok(pi(X)) 41.72/12.37 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 41.72/12.37 times(ok(X1), ok(X2)) -> ok(times(X1, X2)) 41.72/12.37 square(ok(X)) -> ok(square(X)) 41.72/12.37 top(mark(X)) -> top(proper(X)) 41.72/12.37 top(ok(X)) -> top(active(X)) 41.72/12.37 41.72/12.37 S is empty. 41.72/12.37 Rewrite Strategy: FULL 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 41.72/12.37 transformed relative TRS to TRS 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (4) 41.72/12.37 Obligation: 41.72/12.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 41.72/12.37 41.72/12.37 41.72/12.37 The TRS R consists of the following rules: 41.72/12.37 41.72/12.37 s(mark(X)) -> mark(s(X)) 41.72/12.37 posrecip(mark(X)) -> mark(posrecip(X)) 41.72/12.37 negrecip(mark(X)) -> mark(negrecip(X)) 41.72/12.37 cons(mark(X1), X2) -> mark(cons(X1, X2)) 41.72/12.37 rcons(mark(X1), X2) -> mark(rcons(X1, X2)) 41.72/12.37 rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) 41.72/12.37 from(mark(X)) -> mark(from(X)) 41.72/12.37 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 41.72/12.37 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) 41.72/12.37 pi(mark(X)) -> mark(pi(X)) 41.72/12.37 plus(mark(X1), X2) -> mark(plus(X1, X2)) 41.72/12.37 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 41.72/12.37 times(mark(X1), X2) -> mark(times(X1, X2)) 41.72/12.37 times(X1, mark(X2)) -> mark(times(X1, X2)) 41.72/12.37 square(mark(X)) -> mark(square(X)) 41.72/12.37 proper(0) -> ok(0) 41.72/12.37 proper(nil) -> ok(nil) 41.72/12.37 proper(rnil) -> ok(rnil) 41.72/12.37 s(ok(X)) -> ok(s(X)) 41.72/12.37 posrecip(ok(X)) -> ok(posrecip(X)) 41.72/12.37 negrecip(ok(X)) -> ok(negrecip(X)) 41.72/12.37 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 41.72/12.37 rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) 41.72/12.37 from(ok(X)) -> ok(from(X)) 41.72/12.37 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) 41.72/12.37 pi(ok(X)) -> ok(pi(X)) 41.72/12.37 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 41.72/12.37 times(ok(X1), ok(X2)) -> ok(times(X1, X2)) 41.72/12.37 square(ok(X)) -> ok(square(X)) 41.72/12.37 top(mark(X)) -> top(proper(X)) 41.72/12.37 top(ok(X)) -> top(active(X)) 41.72/12.37 41.72/12.37 S is empty. 41.72/12.37 Rewrite Strategy: FULL 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (5) CpxTrsMatchBoundsTAProof (FINISHED) 41.72/12.37 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. 41.72/12.37 41.72/12.37 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 41.72/12.37 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] 41.72/12.37 transitions: 41.72/12.37 mark0(0) -> 0 41.72/12.37 00() -> 0 41.72/12.37 ok0(0) -> 0 41.72/12.37 nil0() -> 0 41.72/12.37 rnil0() -> 0 41.72/12.37 active0(0) -> 0 41.72/12.37 s0(0) -> 1 41.72/12.37 posrecip0(0) -> 2 41.72/12.37 negrecip0(0) -> 3 41.72/12.37 cons0(0, 0) -> 4 41.72/12.37 rcons0(0, 0) -> 5 41.72/12.37 from0(0) -> 6 41.72/12.37 2ndspos0(0, 0) -> 7 41.72/12.37 2ndsneg0(0, 0) -> 8 41.72/12.37 pi0(0) -> 9 41.72/12.37 plus0(0, 0) -> 10 41.72/12.37 times0(0, 0) -> 11 41.72/12.37 square0(0) -> 12 41.72/12.37 proper0(0) -> 13 41.72/12.37 top0(0) -> 14 41.72/12.37 s1(0) -> 15 41.72/12.37 mark1(15) -> 1 41.72/12.37 posrecip1(0) -> 16 41.72/12.37 mark1(16) -> 2 41.72/12.37 negrecip1(0) -> 17 41.72/12.37 mark1(17) -> 3 41.72/12.37 cons1(0, 0) -> 18 41.72/12.37 mark1(18) -> 4 41.72/12.37 rcons1(0, 0) -> 19 41.72/12.37 mark1(19) -> 5 41.72/12.37 from1(0) -> 20 41.72/12.37 mark1(20) -> 6 41.72/12.37 2ndspos1(0, 0) -> 21 41.72/12.37 mark1(21) -> 7 41.72/12.37 2ndsneg1(0, 0) -> 22 41.72/12.37 mark1(22) -> 8 41.72/12.37 pi1(0) -> 23 41.72/12.37 mark1(23) -> 9 41.72/12.37 plus1(0, 0) -> 24 41.72/12.37 mark1(24) -> 10 41.72/12.37 times1(0, 0) -> 25 41.72/12.37 mark1(25) -> 11 41.72/12.37 square1(0) -> 26 41.72/12.37 mark1(26) -> 12 41.72/12.37 01() -> 27 41.72/12.37 ok1(27) -> 13 41.72/12.37 nil1() -> 28 41.72/12.37 ok1(28) -> 13 41.72/12.37 rnil1() -> 29 41.72/12.37 ok1(29) -> 13 41.72/12.37 s1(0) -> 30 41.72/12.37 ok1(30) -> 1 41.72/12.37 posrecip1(0) -> 31 41.72/12.37 ok1(31) -> 2 41.72/12.37 negrecip1(0) -> 32 41.72/12.37 ok1(32) -> 3 41.72/12.37 cons1(0, 0) -> 33 41.72/12.37 ok1(33) -> 4 41.72/12.37 rcons1(0, 0) -> 34 41.72/12.37 ok1(34) -> 5 41.72/12.37 from1(0) -> 35 41.72/12.37 ok1(35) -> 6 41.72/12.37 2ndspos1(0, 0) -> 36 41.72/12.37 ok1(36) -> 7 41.72/12.37 2ndsneg1(0, 0) -> 37 41.72/12.37 ok1(37) -> 8 41.72/12.37 pi1(0) -> 38 41.72/12.37 ok1(38) -> 9 41.72/12.37 plus1(0, 0) -> 39 41.72/12.37 ok1(39) -> 10 41.72/12.37 times1(0, 0) -> 40 41.72/12.37 ok1(40) -> 11 41.72/12.37 square1(0) -> 41 41.72/12.37 ok1(41) -> 12 41.72/12.37 proper1(0) -> 42 41.72/12.37 top1(42) -> 14 41.72/12.37 active1(0) -> 43 41.72/12.37 top1(43) -> 14 41.72/12.37 mark1(15) -> 15 41.72/12.37 mark1(15) -> 30 41.72/12.37 mark1(16) -> 16 41.72/12.37 mark1(16) -> 31 41.72/12.37 mark1(17) -> 17 41.72/12.37 mark1(17) -> 32 41.72/12.37 mark1(18) -> 18 41.72/12.37 mark1(18) -> 33 41.72/12.37 mark1(19) -> 19 41.72/12.37 mark1(19) -> 34 41.72/12.37 mark1(20) -> 20 41.72/12.37 mark1(20) -> 35 41.72/12.37 mark1(21) -> 21 41.72/12.37 mark1(21) -> 36 41.72/12.37 mark1(22) -> 22 41.72/12.37 mark1(22) -> 37 41.72/12.37 mark1(23) -> 23 41.72/12.37 mark1(23) -> 38 41.72/12.37 mark1(24) -> 24 41.72/12.37 mark1(24) -> 39 41.72/12.37 mark1(25) -> 25 41.72/12.37 mark1(25) -> 40 41.72/12.37 mark1(26) -> 26 41.72/12.37 mark1(26) -> 41 41.72/12.37 ok1(27) -> 42 41.72/12.37 ok1(28) -> 42 41.72/12.37 ok1(29) -> 42 41.72/12.37 ok1(30) -> 15 41.72/12.37 ok1(30) -> 30 41.72/12.37 ok1(31) -> 16 41.72/12.37 ok1(31) -> 31 41.72/12.37 ok1(32) -> 17 41.72/12.37 ok1(32) -> 32 41.72/12.37 ok1(33) -> 18 41.72/12.37 ok1(33) -> 33 41.72/12.37 ok1(34) -> 19 41.72/12.37 ok1(34) -> 34 41.72/12.37 ok1(35) -> 20 41.72/12.37 ok1(35) -> 35 41.72/12.37 ok1(36) -> 21 41.72/12.37 ok1(36) -> 36 41.72/12.37 ok1(37) -> 22 41.72/12.37 ok1(37) -> 37 41.72/12.37 ok1(38) -> 23 41.72/12.37 ok1(38) -> 38 41.72/12.37 ok1(39) -> 24 41.72/12.37 ok1(39) -> 39 41.72/12.37 ok1(40) -> 25 41.72/12.37 ok1(40) -> 40 41.72/12.37 ok1(41) -> 26 41.72/12.37 ok1(41) -> 41 41.72/12.37 active2(27) -> 44 41.72/12.37 top2(44) -> 14 41.72/12.37 active2(28) -> 44 41.72/12.37 active2(29) -> 44 41.72/12.37 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (6) 41.72/12.37 BOUNDS(1, n^1) 41.72/12.37 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 41.72/12.37 Transformed a relative TRS into a decreasing-loop problem. 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (8) 41.72/12.37 Obligation: 41.72/12.37 Analyzing the following TRS for decreasing loops: 41.72/12.37 41.72/12.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.72/12.37 41.72/12.37 41.72/12.37 The TRS R consists of the following rules: 41.72/12.37 41.72/12.37 active(from(X)) -> mark(cons(X, from(s(X)))) 41.72/12.37 active(2ndspos(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) 41.72/12.37 active(2ndsneg(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) 41.72/12.37 active(pi(X)) -> mark(2ndspos(X, from(0))) 41.72/12.37 active(plus(0, Y)) -> mark(Y) 41.72/12.37 active(plus(s(X), Y)) -> mark(s(plus(X, Y))) 41.72/12.37 active(times(0, Y)) -> mark(0) 41.72/12.37 active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) 41.72/12.37 active(square(X)) -> mark(times(X, X)) 41.72/12.37 active(s(X)) -> s(active(X)) 41.72/12.37 active(posrecip(X)) -> posrecip(active(X)) 41.72/12.37 active(negrecip(X)) -> negrecip(active(X)) 41.72/12.37 active(cons(X1, X2)) -> cons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(X1, active(X2)) 41.72/12.37 active(from(X)) -> from(active(X)) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) 41.72/12.37 active(pi(X)) -> pi(active(X)) 41.72/12.37 active(plus(X1, X2)) -> plus(active(X1), X2) 41.72/12.37 active(plus(X1, X2)) -> plus(X1, active(X2)) 41.72/12.37 active(times(X1, X2)) -> times(active(X1), X2) 41.72/12.37 active(times(X1, X2)) -> times(X1, active(X2)) 41.72/12.37 active(square(X)) -> square(active(X)) 41.72/12.37 s(mark(X)) -> mark(s(X)) 41.72/12.37 posrecip(mark(X)) -> mark(posrecip(X)) 41.72/12.37 negrecip(mark(X)) -> mark(negrecip(X)) 41.72/12.37 cons(mark(X1), X2) -> mark(cons(X1, X2)) 41.72/12.37 rcons(mark(X1), X2) -> mark(rcons(X1, X2)) 41.72/12.37 rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) 41.72/12.37 from(mark(X)) -> mark(from(X)) 41.72/12.37 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 41.72/12.37 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) 41.72/12.37 pi(mark(X)) -> mark(pi(X)) 41.72/12.37 plus(mark(X1), X2) -> mark(plus(X1, X2)) 41.72/12.37 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 41.72/12.37 times(mark(X1), X2) -> mark(times(X1, X2)) 41.72/12.37 times(X1, mark(X2)) -> mark(times(X1, X2)) 41.72/12.37 square(mark(X)) -> mark(square(X)) 41.72/12.37 proper(0) -> ok(0) 41.72/12.37 proper(s(X)) -> s(proper(X)) 41.72/12.37 proper(posrecip(X)) -> posrecip(proper(X)) 41.72/12.37 proper(negrecip(X)) -> negrecip(proper(X)) 41.72/12.37 proper(nil) -> ok(nil) 41.72/12.37 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 41.72/12.37 proper(rnil) -> ok(rnil) 41.72/12.37 proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) 41.72/12.37 proper(from(X)) -> from(proper(X)) 41.72/12.37 proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) 41.72/12.37 proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) 41.72/12.37 proper(pi(X)) -> pi(proper(X)) 41.72/12.37 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 41.72/12.37 proper(times(X1, X2)) -> times(proper(X1), proper(X2)) 41.72/12.37 proper(square(X)) -> square(proper(X)) 41.72/12.37 s(ok(X)) -> ok(s(X)) 41.72/12.37 posrecip(ok(X)) -> ok(posrecip(X)) 41.72/12.37 negrecip(ok(X)) -> ok(negrecip(X)) 41.72/12.37 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 41.72/12.37 rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) 41.72/12.37 from(ok(X)) -> ok(from(X)) 41.72/12.37 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) 41.72/12.37 pi(ok(X)) -> ok(pi(X)) 41.72/12.37 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 41.72/12.37 times(ok(X1), ok(X2)) -> ok(times(X1, X2)) 41.72/12.37 square(ok(X)) -> ok(square(X)) 41.72/12.37 top(mark(X)) -> top(proper(X)) 41.72/12.37 top(ok(X)) -> top(active(X)) 41.72/12.37 41.72/12.37 S is empty. 41.72/12.37 Rewrite Strategy: FULL 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (9) DecreasingLoopProof (LOWER BOUND(ID)) 41.72/12.37 The following loop(s) give(s) rise to the lower bound Omega(n^1): 41.72/12.37 41.72/12.37 The rewrite sequence 41.72/12.37 41.72/12.37 times(ok(X1), ok(X2)) ->^+ ok(times(X1, X2)) 41.72/12.37 41.72/12.37 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 41.72/12.37 41.72/12.37 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 41.72/12.37 41.72/12.37 The result substitution is [ ]. 41.72/12.37 41.72/12.37 41.72/12.37 41.72/12.37 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (10) 41.72/12.37 Complex Obligation (BEST) 41.72/12.37 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (11) 41.72/12.37 Obligation: 41.72/12.37 Proved the lower bound n^1 for the following obligation: 41.72/12.37 41.72/12.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.72/12.37 41.72/12.37 41.72/12.37 The TRS R consists of the following rules: 41.72/12.37 41.72/12.37 active(from(X)) -> mark(cons(X, from(s(X)))) 41.72/12.37 active(2ndspos(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) 41.72/12.37 active(2ndsneg(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) 41.72/12.37 active(pi(X)) -> mark(2ndspos(X, from(0))) 41.72/12.37 active(plus(0, Y)) -> mark(Y) 41.72/12.37 active(plus(s(X), Y)) -> mark(s(plus(X, Y))) 41.72/12.37 active(times(0, Y)) -> mark(0) 41.72/12.37 active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) 41.72/12.37 active(square(X)) -> mark(times(X, X)) 41.72/12.37 active(s(X)) -> s(active(X)) 41.72/12.37 active(posrecip(X)) -> posrecip(active(X)) 41.72/12.37 active(negrecip(X)) -> negrecip(active(X)) 41.72/12.37 active(cons(X1, X2)) -> cons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(X1, active(X2)) 41.72/12.37 active(from(X)) -> from(active(X)) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) 41.72/12.37 active(pi(X)) -> pi(active(X)) 41.72/12.37 active(plus(X1, X2)) -> plus(active(X1), X2) 41.72/12.37 active(plus(X1, X2)) -> plus(X1, active(X2)) 41.72/12.37 active(times(X1, X2)) -> times(active(X1), X2) 41.72/12.37 active(times(X1, X2)) -> times(X1, active(X2)) 41.72/12.37 active(square(X)) -> square(active(X)) 41.72/12.37 s(mark(X)) -> mark(s(X)) 41.72/12.37 posrecip(mark(X)) -> mark(posrecip(X)) 41.72/12.37 negrecip(mark(X)) -> mark(negrecip(X)) 41.72/12.37 cons(mark(X1), X2) -> mark(cons(X1, X2)) 41.72/12.37 rcons(mark(X1), X2) -> mark(rcons(X1, X2)) 41.72/12.37 rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) 41.72/12.37 from(mark(X)) -> mark(from(X)) 41.72/12.37 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 41.72/12.37 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) 41.72/12.37 pi(mark(X)) -> mark(pi(X)) 41.72/12.37 plus(mark(X1), X2) -> mark(plus(X1, X2)) 41.72/12.37 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 41.72/12.37 times(mark(X1), X2) -> mark(times(X1, X2)) 41.72/12.37 times(X1, mark(X2)) -> mark(times(X1, X2)) 41.72/12.37 square(mark(X)) -> mark(square(X)) 41.72/12.37 proper(0) -> ok(0) 41.72/12.37 proper(s(X)) -> s(proper(X)) 41.72/12.37 proper(posrecip(X)) -> posrecip(proper(X)) 41.72/12.37 proper(negrecip(X)) -> negrecip(proper(X)) 41.72/12.37 proper(nil) -> ok(nil) 41.72/12.37 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 41.72/12.37 proper(rnil) -> ok(rnil) 41.72/12.37 proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) 41.72/12.37 proper(from(X)) -> from(proper(X)) 41.72/12.37 proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) 41.72/12.37 proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) 41.72/12.37 proper(pi(X)) -> pi(proper(X)) 41.72/12.37 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 41.72/12.37 proper(times(X1, X2)) -> times(proper(X1), proper(X2)) 41.72/12.37 proper(square(X)) -> square(proper(X)) 41.72/12.37 s(ok(X)) -> ok(s(X)) 41.72/12.37 posrecip(ok(X)) -> ok(posrecip(X)) 41.72/12.37 negrecip(ok(X)) -> ok(negrecip(X)) 41.72/12.37 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 41.72/12.37 rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) 41.72/12.37 from(ok(X)) -> ok(from(X)) 41.72/12.37 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) 41.72/12.37 pi(ok(X)) -> ok(pi(X)) 41.72/12.37 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 41.72/12.37 times(ok(X1), ok(X2)) -> ok(times(X1, X2)) 41.72/12.37 square(ok(X)) -> ok(square(X)) 41.72/12.37 top(mark(X)) -> top(proper(X)) 41.72/12.37 top(ok(X)) -> top(active(X)) 41.72/12.37 41.72/12.37 S is empty. 41.72/12.37 Rewrite Strategy: FULL 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (12) LowerBoundPropagationProof (FINISHED) 41.72/12.37 Propagated lower bound. 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (13) 41.72/12.37 BOUNDS(n^1, INF) 41.72/12.37 41.72/12.37 ---------------------------------------- 41.72/12.37 41.72/12.37 (14) 41.72/12.37 Obligation: 41.72/12.37 Analyzing the following TRS for decreasing loops: 41.72/12.37 41.72/12.37 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 41.72/12.37 41.72/12.37 41.72/12.37 The TRS R consists of the following rules: 41.72/12.37 41.72/12.37 active(from(X)) -> mark(cons(X, from(s(X)))) 41.72/12.37 active(2ndspos(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndspos(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(posrecip(Y), 2ndsneg(N, Z))) 41.72/12.37 active(2ndsneg(0, Z)) -> mark(rnil) 41.72/12.37 active(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> mark(rcons(negrecip(Y), 2ndspos(N, Z))) 41.72/12.37 active(pi(X)) -> mark(2ndspos(X, from(0))) 41.72/12.37 active(plus(0, Y)) -> mark(Y) 41.72/12.37 active(plus(s(X), Y)) -> mark(s(plus(X, Y))) 41.72/12.37 active(times(0, Y)) -> mark(0) 41.72/12.37 active(times(s(X), Y)) -> mark(plus(Y, times(X, Y))) 41.72/12.37 active(square(X)) -> mark(times(X, X)) 41.72/12.37 active(s(X)) -> s(active(X)) 41.72/12.37 active(posrecip(X)) -> posrecip(active(X)) 41.72/12.37 active(negrecip(X)) -> negrecip(active(X)) 41.72/12.37 active(cons(X1, X2)) -> cons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(active(X1), X2) 41.72/12.37 active(rcons(X1, X2)) -> rcons(X1, active(X2)) 41.72/12.37 active(from(X)) -> from(active(X)) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(active(X1), X2) 41.72/12.37 active(2ndspos(X1, X2)) -> 2ndspos(X1, active(X2)) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(active(X1), X2) 41.72/12.37 active(2ndsneg(X1, X2)) -> 2ndsneg(X1, active(X2)) 41.72/12.37 active(pi(X)) -> pi(active(X)) 41.72/12.37 active(plus(X1, X2)) -> plus(active(X1), X2) 41.72/12.37 active(plus(X1, X2)) -> plus(X1, active(X2)) 41.72/12.37 active(times(X1, X2)) -> times(active(X1), X2) 41.72/12.37 active(times(X1, X2)) -> times(X1, active(X2)) 41.72/12.37 active(square(X)) -> square(active(X)) 41.72/12.37 s(mark(X)) -> mark(s(X)) 41.72/12.37 posrecip(mark(X)) -> mark(posrecip(X)) 41.72/12.37 negrecip(mark(X)) -> mark(negrecip(X)) 41.72/12.37 cons(mark(X1), X2) -> mark(cons(X1, X2)) 41.72/12.37 rcons(mark(X1), X2) -> mark(rcons(X1, X2)) 41.72/12.37 rcons(X1, mark(X2)) -> mark(rcons(X1, X2)) 41.72/12.37 from(mark(X)) -> mark(from(X)) 41.72/12.37 2ndspos(mark(X1), X2) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndspos(X1, mark(X2)) -> mark(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(mark(X1), X2) -> mark(2ndsneg(X1, X2)) 41.72/12.37 2ndsneg(X1, mark(X2)) -> mark(2ndsneg(X1, X2)) 41.72/12.37 pi(mark(X)) -> mark(pi(X)) 41.72/12.37 plus(mark(X1), X2) -> mark(plus(X1, X2)) 41.72/12.37 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 41.72/12.37 times(mark(X1), X2) -> mark(times(X1, X2)) 41.72/12.37 times(X1, mark(X2)) -> mark(times(X1, X2)) 41.72/12.37 square(mark(X)) -> mark(square(X)) 41.72/12.37 proper(0) -> ok(0) 41.72/12.37 proper(s(X)) -> s(proper(X)) 41.72/12.37 proper(posrecip(X)) -> posrecip(proper(X)) 41.72/12.37 proper(negrecip(X)) -> negrecip(proper(X)) 41.72/12.37 proper(nil) -> ok(nil) 41.72/12.37 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 41.72/12.37 proper(rnil) -> ok(rnil) 41.72/12.37 proper(rcons(X1, X2)) -> rcons(proper(X1), proper(X2)) 41.72/12.37 proper(from(X)) -> from(proper(X)) 41.72/12.37 proper(2ndspos(X1, X2)) -> 2ndspos(proper(X1), proper(X2)) 41.72/12.37 proper(2ndsneg(X1, X2)) -> 2ndsneg(proper(X1), proper(X2)) 41.72/12.37 proper(pi(X)) -> pi(proper(X)) 41.72/12.37 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 41.72/12.37 proper(times(X1, X2)) -> times(proper(X1), proper(X2)) 41.72/12.37 proper(square(X)) -> square(proper(X)) 41.72/12.37 s(ok(X)) -> ok(s(X)) 41.72/12.37 posrecip(ok(X)) -> ok(posrecip(X)) 41.72/12.37 negrecip(ok(X)) -> ok(negrecip(X)) 41.72/12.37 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 41.72/12.37 rcons(ok(X1), ok(X2)) -> ok(rcons(X1, X2)) 41.72/12.37 from(ok(X)) -> ok(from(X)) 41.72/12.37 2ndspos(ok(X1), ok(X2)) -> ok(2ndspos(X1, X2)) 41.72/12.37 2ndsneg(ok(X1), ok(X2)) -> ok(2ndsneg(X1, X2)) 41.72/12.37 pi(ok(X)) -> ok(pi(X)) 41.72/12.37 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 41.72/12.37 times(ok(X1), ok(X2)) -> ok(times(X1, X2)) 41.72/12.37 square(ok(X)) -> ok(square(X)) 41.72/12.37 top(mark(X)) -> top(proper(X)) 41.72/12.37 top(ok(X)) -> top(active(X)) 41.72/12.37 41.72/12.37 S is empty. 41.72/12.37 Rewrite Strategy: FULL 42.00/12.42 EOF