27.35/8.25 WORST_CASE(Omega(n^1), O(n^1)) 27.35/8.27 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 27.35/8.27 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 27.35/8.27 27.35/8.27 27.35/8.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.35/8.27 27.35/8.27 (0) CpxTRS 27.35/8.27 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 27.35/8.27 (2) CpxTRS 27.35/8.27 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 27.35/8.27 (4) CpxTRS 27.35/8.27 (5) CpxTrsMatchBoundsTAProof [FINISHED, 66 ms] 27.35/8.27 (6) BOUNDS(1, n^1) 27.35/8.27 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 27.35/8.27 (8) TRS for Loop Detection 27.35/8.27 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 27.35/8.27 (10) BEST 27.35/8.27 (11) proven lower bound 27.35/8.27 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 27.35/8.27 (13) BOUNDS(n^1, INF) 27.35/8.27 (14) TRS for Loop Detection 27.35/8.27 27.35/8.27 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (0) 27.35/8.27 Obligation: 27.35/8.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.35/8.27 27.35/8.27 27.35/8.27 The TRS R consists of the following rules: 27.35/8.27 27.35/8.27 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 27.35/8.27 active(sqr(0)) -> mark(0) 27.35/8.27 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 27.35/8.27 active(dbl(0)) -> mark(0) 27.35/8.27 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 27.35/8.27 active(add(0, X)) -> mark(X) 27.35/8.27 active(add(s(X), Y)) -> mark(s(add(X, Y))) 27.35/8.27 active(first(0, X)) -> mark(nil) 27.35/8.27 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 27.35/8.27 active(half(0)) -> mark(0) 27.35/8.27 active(half(s(0))) -> mark(0) 27.35/8.27 active(half(s(s(X)))) -> mark(s(half(X))) 27.35/8.27 active(half(dbl(X))) -> mark(X) 27.35/8.27 active(terms(X)) -> terms(active(X)) 27.35/8.27 active(cons(X1, X2)) -> cons(active(X1), X2) 27.35/8.27 active(recip(X)) -> recip(active(X)) 27.35/8.27 active(sqr(X)) -> sqr(active(X)) 27.35/8.27 active(s(X)) -> s(active(X)) 27.35/8.27 active(add(X1, X2)) -> add(active(X1), X2) 27.35/8.27 active(add(X1, X2)) -> add(X1, active(X2)) 27.35/8.27 active(dbl(X)) -> dbl(active(X)) 27.35/8.27 active(first(X1, X2)) -> first(active(X1), X2) 27.35/8.27 active(first(X1, X2)) -> first(X1, active(X2)) 27.35/8.27 active(half(X)) -> half(active(X)) 27.35/8.27 terms(mark(X)) -> mark(terms(X)) 27.35/8.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 27.35/8.27 recip(mark(X)) -> mark(recip(X)) 27.35/8.27 sqr(mark(X)) -> mark(sqr(X)) 27.35/8.27 s(mark(X)) -> mark(s(X)) 27.35/8.27 add(mark(X1), X2) -> mark(add(X1, X2)) 27.35/8.27 add(X1, mark(X2)) -> mark(add(X1, X2)) 27.35/8.27 dbl(mark(X)) -> mark(dbl(X)) 27.35/8.27 first(mark(X1), X2) -> mark(first(X1, X2)) 27.35/8.27 first(X1, mark(X2)) -> mark(first(X1, X2)) 27.35/8.27 half(mark(X)) -> mark(half(X)) 27.35/8.27 proper(terms(X)) -> terms(proper(X)) 27.35/8.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 27.35/8.27 proper(recip(X)) -> recip(proper(X)) 27.35/8.27 proper(sqr(X)) -> sqr(proper(X)) 27.35/8.27 proper(s(X)) -> s(proper(X)) 27.35/8.27 proper(0) -> ok(0) 27.35/8.27 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 27.35/8.27 proper(dbl(X)) -> dbl(proper(X)) 27.35/8.27 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 27.35/8.27 proper(nil) -> ok(nil) 27.35/8.27 proper(half(X)) -> half(proper(X)) 27.35/8.27 terms(ok(X)) -> ok(terms(X)) 27.35/8.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 27.35/8.27 recip(ok(X)) -> ok(recip(X)) 27.35/8.27 sqr(ok(X)) -> ok(sqr(X)) 27.35/8.27 s(ok(X)) -> ok(s(X)) 27.35/8.27 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 27.35/8.27 dbl(ok(X)) -> ok(dbl(X)) 27.35/8.27 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 27.35/8.27 half(ok(X)) -> ok(half(X)) 27.35/8.27 top(mark(X)) -> top(proper(X)) 27.35/8.27 top(ok(X)) -> top(active(X)) 27.35/8.27 27.35/8.27 S is empty. 27.35/8.27 Rewrite Strategy: FULL 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 27.35/8.27 The following defined symbols can occur below the 0th argument of top: proper, active 27.35/8.27 The following defined symbols can occur below the 0th argument of proper: proper, active 27.35/8.27 The following defined symbols can occur below the 0th argument of active: proper, active 27.35/8.27 27.35/8.27 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 27.35/8.27 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 27.35/8.27 active(sqr(0)) -> mark(0) 27.35/8.27 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 27.35/8.27 active(dbl(0)) -> mark(0) 27.35/8.27 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 27.35/8.27 active(add(0, X)) -> mark(X) 27.35/8.27 active(add(s(X), Y)) -> mark(s(add(X, Y))) 27.35/8.27 active(first(0, X)) -> mark(nil) 27.35/8.27 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 27.35/8.27 active(half(0)) -> mark(0) 27.35/8.27 active(half(s(0))) -> mark(0) 27.35/8.27 active(half(s(s(X)))) -> mark(s(half(X))) 27.35/8.27 active(half(dbl(X))) -> mark(X) 27.35/8.27 active(terms(X)) -> terms(active(X)) 27.35/8.27 active(cons(X1, X2)) -> cons(active(X1), X2) 27.35/8.27 active(recip(X)) -> recip(active(X)) 27.35/8.27 active(sqr(X)) -> sqr(active(X)) 27.35/8.27 active(s(X)) -> s(active(X)) 27.35/8.27 active(add(X1, X2)) -> add(active(X1), X2) 27.35/8.27 active(add(X1, X2)) -> add(X1, active(X2)) 27.35/8.27 active(dbl(X)) -> dbl(active(X)) 27.35/8.27 active(first(X1, X2)) -> first(active(X1), X2) 27.35/8.27 active(first(X1, X2)) -> first(X1, active(X2)) 27.35/8.27 active(half(X)) -> half(active(X)) 27.35/8.27 proper(terms(X)) -> terms(proper(X)) 27.35/8.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 27.35/8.27 proper(recip(X)) -> recip(proper(X)) 27.35/8.27 proper(sqr(X)) -> sqr(proper(X)) 27.35/8.27 proper(s(X)) -> s(proper(X)) 27.35/8.27 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 27.35/8.27 proper(dbl(X)) -> dbl(proper(X)) 27.35/8.27 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 27.35/8.27 proper(half(X)) -> half(proper(X)) 27.35/8.27 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (2) 27.35/8.27 Obligation: 27.35/8.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 27.35/8.27 27.35/8.27 27.35/8.27 The TRS R consists of the following rules: 27.35/8.27 27.35/8.27 terms(mark(X)) -> mark(terms(X)) 27.35/8.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 27.35/8.27 recip(mark(X)) -> mark(recip(X)) 27.35/8.27 sqr(mark(X)) -> mark(sqr(X)) 27.35/8.27 s(mark(X)) -> mark(s(X)) 27.35/8.27 add(mark(X1), X2) -> mark(add(X1, X2)) 27.35/8.27 add(X1, mark(X2)) -> mark(add(X1, X2)) 27.35/8.27 dbl(mark(X)) -> mark(dbl(X)) 27.35/8.27 first(mark(X1), X2) -> mark(first(X1, X2)) 27.35/8.27 first(X1, mark(X2)) -> mark(first(X1, X2)) 27.35/8.27 half(mark(X)) -> mark(half(X)) 27.35/8.27 proper(0) -> ok(0) 27.35/8.27 proper(nil) -> ok(nil) 27.35/8.27 terms(ok(X)) -> ok(terms(X)) 27.35/8.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 27.35/8.27 recip(ok(X)) -> ok(recip(X)) 27.35/8.27 sqr(ok(X)) -> ok(sqr(X)) 27.35/8.27 s(ok(X)) -> ok(s(X)) 27.35/8.27 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 27.35/8.27 dbl(ok(X)) -> ok(dbl(X)) 27.35/8.27 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 27.35/8.27 half(ok(X)) -> ok(half(X)) 27.35/8.27 top(mark(X)) -> top(proper(X)) 27.35/8.27 top(ok(X)) -> top(active(X)) 27.35/8.27 27.35/8.27 S is empty. 27.35/8.27 Rewrite Strategy: FULL 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 27.35/8.27 transformed relative TRS to TRS 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (4) 27.35/8.27 Obligation: 27.35/8.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 27.35/8.27 27.35/8.27 27.35/8.27 The TRS R consists of the following rules: 27.35/8.27 27.35/8.27 terms(mark(X)) -> mark(terms(X)) 27.35/8.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 27.35/8.27 recip(mark(X)) -> mark(recip(X)) 27.35/8.27 sqr(mark(X)) -> mark(sqr(X)) 27.35/8.27 s(mark(X)) -> mark(s(X)) 27.35/8.27 add(mark(X1), X2) -> mark(add(X1, X2)) 27.35/8.27 add(X1, mark(X2)) -> mark(add(X1, X2)) 27.35/8.27 dbl(mark(X)) -> mark(dbl(X)) 27.35/8.27 first(mark(X1), X2) -> mark(first(X1, X2)) 27.35/8.27 first(X1, mark(X2)) -> mark(first(X1, X2)) 27.35/8.27 half(mark(X)) -> mark(half(X)) 27.35/8.27 proper(0) -> ok(0) 27.35/8.27 proper(nil) -> ok(nil) 27.35/8.27 terms(ok(X)) -> ok(terms(X)) 27.35/8.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 27.35/8.27 recip(ok(X)) -> ok(recip(X)) 27.35/8.27 sqr(ok(X)) -> ok(sqr(X)) 27.35/8.27 s(ok(X)) -> ok(s(X)) 27.35/8.27 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 27.35/8.27 dbl(ok(X)) -> ok(dbl(X)) 27.35/8.27 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 27.35/8.27 half(ok(X)) -> ok(half(X)) 27.35/8.27 top(mark(X)) -> top(proper(X)) 27.35/8.27 top(ok(X)) -> top(active(X)) 27.35/8.27 27.35/8.27 S is empty. 27.35/8.27 Rewrite Strategy: FULL 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (5) CpxTrsMatchBoundsTAProof (FINISHED) 27.35/8.27 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. 27.35/8.27 27.35/8.27 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 27.35/8.27 final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] 27.35/8.27 transitions: 27.35/8.27 mark0(0) -> 0 27.35/8.27 00() -> 0 27.35/8.27 ok0(0) -> 0 27.35/8.27 nil0() -> 0 27.35/8.27 active0(0) -> 0 27.35/8.27 terms0(0) -> 1 27.35/8.27 cons0(0, 0) -> 2 27.35/8.27 recip0(0) -> 3 27.35/8.27 sqr0(0) -> 4 27.35/8.27 s0(0) -> 5 27.35/8.27 add0(0, 0) -> 6 27.35/8.27 dbl0(0) -> 7 27.35/8.27 first0(0, 0) -> 8 27.35/8.27 half0(0) -> 9 27.35/8.27 proper0(0) -> 10 27.35/8.27 top0(0) -> 11 27.35/8.27 terms1(0) -> 12 27.35/8.27 mark1(12) -> 1 27.35/8.27 cons1(0, 0) -> 13 27.35/8.27 mark1(13) -> 2 27.35/8.27 recip1(0) -> 14 27.35/8.27 mark1(14) -> 3 27.35/8.27 sqr1(0) -> 15 27.35/8.27 mark1(15) -> 4 27.35/8.27 s1(0) -> 16 27.35/8.27 mark1(16) -> 5 27.35/8.27 add1(0, 0) -> 17 27.35/8.27 mark1(17) -> 6 27.35/8.27 dbl1(0) -> 18 27.35/8.27 mark1(18) -> 7 27.35/8.27 first1(0, 0) -> 19 27.35/8.27 mark1(19) -> 8 27.35/8.27 half1(0) -> 20 27.35/8.27 mark1(20) -> 9 27.35/8.27 01() -> 21 27.35/8.27 ok1(21) -> 10 27.35/8.27 nil1() -> 22 27.35/8.27 ok1(22) -> 10 27.35/8.27 terms1(0) -> 23 27.35/8.27 ok1(23) -> 1 27.35/8.27 cons1(0, 0) -> 24 27.35/8.27 ok1(24) -> 2 27.35/8.27 recip1(0) -> 25 27.35/8.27 ok1(25) -> 3 27.35/8.27 sqr1(0) -> 26 27.35/8.27 ok1(26) -> 4 27.35/8.27 s1(0) -> 27 27.35/8.27 ok1(27) -> 5 27.35/8.27 add1(0, 0) -> 28 27.35/8.27 ok1(28) -> 6 27.35/8.27 dbl1(0) -> 29 27.35/8.27 ok1(29) -> 7 27.35/8.27 first1(0, 0) -> 30 27.35/8.27 ok1(30) -> 8 27.35/8.27 half1(0) -> 31 27.35/8.27 ok1(31) -> 9 27.35/8.27 proper1(0) -> 32 27.35/8.27 top1(32) -> 11 27.35/8.27 active1(0) -> 33 27.35/8.27 top1(33) -> 11 27.35/8.27 mark1(12) -> 12 27.35/8.27 mark1(12) -> 23 27.35/8.27 mark1(13) -> 13 27.35/8.27 mark1(13) -> 24 27.35/8.27 mark1(14) -> 14 27.35/8.27 mark1(14) -> 25 27.35/8.27 mark1(15) -> 15 27.35/8.27 mark1(15) -> 26 27.35/8.27 mark1(16) -> 16 27.35/8.27 mark1(16) -> 27 27.35/8.27 mark1(17) -> 17 27.35/8.27 mark1(17) -> 28 27.35/8.27 mark1(18) -> 18 27.35/8.27 mark1(18) -> 29 27.35/8.27 mark1(19) -> 19 27.35/8.27 mark1(19) -> 30 27.35/8.27 mark1(20) -> 20 27.35/8.27 mark1(20) -> 31 27.35/8.27 ok1(21) -> 32 27.35/8.27 ok1(22) -> 32 27.35/8.27 ok1(23) -> 12 27.35/8.27 ok1(23) -> 23 27.35/8.27 ok1(24) -> 13 27.35/8.27 ok1(24) -> 24 27.35/8.27 ok1(25) -> 14 27.35/8.27 ok1(25) -> 25 27.35/8.27 ok1(26) -> 15 27.35/8.27 ok1(26) -> 26 27.35/8.27 ok1(27) -> 16 27.35/8.27 ok1(27) -> 27 27.35/8.27 ok1(28) -> 17 27.35/8.27 ok1(28) -> 28 27.35/8.27 ok1(29) -> 18 27.35/8.27 ok1(29) -> 29 27.35/8.27 ok1(30) -> 19 27.35/8.27 ok1(30) -> 30 27.35/8.27 ok1(31) -> 20 27.35/8.27 ok1(31) -> 31 27.35/8.27 active2(21) -> 34 27.35/8.27 top2(34) -> 11 27.35/8.27 active2(22) -> 34 27.35/8.27 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (6) 27.35/8.27 BOUNDS(1, n^1) 27.35/8.27 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 27.35/8.27 Transformed a relative TRS into a decreasing-loop problem. 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (8) 27.35/8.27 Obligation: 27.35/8.27 Analyzing the following TRS for decreasing loops: 27.35/8.27 27.35/8.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.35/8.27 27.35/8.27 27.35/8.27 The TRS R consists of the following rules: 27.35/8.27 27.35/8.27 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 27.35/8.27 active(sqr(0)) -> mark(0) 27.35/8.27 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 27.35/8.27 active(dbl(0)) -> mark(0) 27.35/8.27 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 27.35/8.27 active(add(0, X)) -> mark(X) 27.35/8.27 active(add(s(X), Y)) -> mark(s(add(X, Y))) 27.35/8.27 active(first(0, X)) -> mark(nil) 27.35/8.27 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 27.35/8.27 active(half(0)) -> mark(0) 27.35/8.27 active(half(s(0))) -> mark(0) 27.35/8.27 active(half(s(s(X)))) -> mark(s(half(X))) 27.35/8.27 active(half(dbl(X))) -> mark(X) 27.35/8.27 active(terms(X)) -> terms(active(X)) 27.35/8.27 active(cons(X1, X2)) -> cons(active(X1), X2) 27.35/8.27 active(recip(X)) -> recip(active(X)) 27.35/8.27 active(sqr(X)) -> sqr(active(X)) 27.35/8.27 active(s(X)) -> s(active(X)) 27.35/8.27 active(add(X1, X2)) -> add(active(X1), X2) 27.35/8.27 active(add(X1, X2)) -> add(X1, active(X2)) 27.35/8.27 active(dbl(X)) -> dbl(active(X)) 27.35/8.27 active(first(X1, X2)) -> first(active(X1), X2) 27.35/8.27 active(first(X1, X2)) -> first(X1, active(X2)) 27.35/8.27 active(half(X)) -> half(active(X)) 27.35/8.27 terms(mark(X)) -> mark(terms(X)) 27.35/8.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 27.35/8.27 recip(mark(X)) -> mark(recip(X)) 27.35/8.27 sqr(mark(X)) -> mark(sqr(X)) 27.35/8.27 s(mark(X)) -> mark(s(X)) 27.35/8.27 add(mark(X1), X2) -> mark(add(X1, X2)) 27.35/8.27 add(X1, mark(X2)) -> mark(add(X1, X2)) 27.35/8.27 dbl(mark(X)) -> mark(dbl(X)) 27.35/8.27 first(mark(X1), X2) -> mark(first(X1, X2)) 27.35/8.27 first(X1, mark(X2)) -> mark(first(X1, X2)) 27.35/8.27 half(mark(X)) -> mark(half(X)) 27.35/8.27 proper(terms(X)) -> terms(proper(X)) 27.35/8.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 27.35/8.27 proper(recip(X)) -> recip(proper(X)) 27.35/8.27 proper(sqr(X)) -> sqr(proper(X)) 27.35/8.27 proper(s(X)) -> s(proper(X)) 27.35/8.27 proper(0) -> ok(0) 27.35/8.27 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 27.35/8.27 proper(dbl(X)) -> dbl(proper(X)) 27.35/8.27 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 27.35/8.27 proper(nil) -> ok(nil) 27.35/8.27 proper(half(X)) -> half(proper(X)) 27.35/8.27 terms(ok(X)) -> ok(terms(X)) 27.35/8.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 27.35/8.27 recip(ok(X)) -> ok(recip(X)) 27.35/8.27 sqr(ok(X)) -> ok(sqr(X)) 27.35/8.27 s(ok(X)) -> ok(s(X)) 27.35/8.27 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 27.35/8.27 dbl(ok(X)) -> ok(dbl(X)) 27.35/8.27 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 27.35/8.27 half(ok(X)) -> ok(half(X)) 27.35/8.27 top(mark(X)) -> top(proper(X)) 27.35/8.27 top(ok(X)) -> top(active(X)) 27.35/8.27 27.35/8.27 S is empty. 27.35/8.27 Rewrite Strategy: FULL 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (9) DecreasingLoopProof (LOWER BOUND(ID)) 27.35/8.27 The following loop(s) give(s) rise to the lower bound Omega(n^1): 27.35/8.27 27.35/8.27 The rewrite sequence 27.35/8.27 27.35/8.27 terms(ok(X)) ->^+ ok(terms(X)) 27.35/8.27 27.35/8.27 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 27.35/8.27 27.35/8.27 The pumping substitution is [X / ok(X)]. 27.35/8.27 27.35/8.27 The result substitution is [ ]. 27.35/8.27 27.35/8.27 27.35/8.27 27.35/8.27 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (10) 27.35/8.27 Complex Obligation (BEST) 27.35/8.27 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (11) 27.35/8.27 Obligation: 27.35/8.27 Proved the lower bound n^1 for the following obligation: 27.35/8.27 27.35/8.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.35/8.27 27.35/8.27 27.35/8.27 The TRS R consists of the following rules: 27.35/8.27 27.35/8.27 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 27.35/8.27 active(sqr(0)) -> mark(0) 27.35/8.27 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 27.35/8.27 active(dbl(0)) -> mark(0) 27.35/8.27 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 27.35/8.27 active(add(0, X)) -> mark(X) 27.35/8.27 active(add(s(X), Y)) -> mark(s(add(X, Y))) 27.35/8.27 active(first(0, X)) -> mark(nil) 27.35/8.27 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 27.35/8.27 active(half(0)) -> mark(0) 27.35/8.27 active(half(s(0))) -> mark(0) 27.35/8.27 active(half(s(s(X)))) -> mark(s(half(X))) 27.35/8.27 active(half(dbl(X))) -> mark(X) 27.35/8.27 active(terms(X)) -> terms(active(X)) 27.35/8.27 active(cons(X1, X2)) -> cons(active(X1), X2) 27.35/8.27 active(recip(X)) -> recip(active(X)) 27.35/8.27 active(sqr(X)) -> sqr(active(X)) 27.35/8.27 active(s(X)) -> s(active(X)) 27.35/8.27 active(add(X1, X2)) -> add(active(X1), X2) 27.35/8.27 active(add(X1, X2)) -> add(X1, active(X2)) 27.35/8.27 active(dbl(X)) -> dbl(active(X)) 27.35/8.27 active(first(X1, X2)) -> first(active(X1), X2) 27.35/8.27 active(first(X1, X2)) -> first(X1, active(X2)) 27.35/8.27 active(half(X)) -> half(active(X)) 27.35/8.27 terms(mark(X)) -> mark(terms(X)) 27.35/8.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 27.35/8.27 recip(mark(X)) -> mark(recip(X)) 27.35/8.27 sqr(mark(X)) -> mark(sqr(X)) 27.35/8.27 s(mark(X)) -> mark(s(X)) 27.35/8.27 add(mark(X1), X2) -> mark(add(X1, X2)) 27.35/8.27 add(X1, mark(X2)) -> mark(add(X1, X2)) 27.35/8.27 dbl(mark(X)) -> mark(dbl(X)) 27.35/8.27 first(mark(X1), X2) -> mark(first(X1, X2)) 27.35/8.27 first(X1, mark(X2)) -> mark(first(X1, X2)) 27.35/8.27 half(mark(X)) -> mark(half(X)) 27.35/8.27 proper(terms(X)) -> terms(proper(X)) 27.35/8.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 27.35/8.27 proper(recip(X)) -> recip(proper(X)) 27.35/8.27 proper(sqr(X)) -> sqr(proper(X)) 27.35/8.27 proper(s(X)) -> s(proper(X)) 27.35/8.27 proper(0) -> ok(0) 27.35/8.27 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 27.35/8.27 proper(dbl(X)) -> dbl(proper(X)) 27.35/8.27 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 27.35/8.27 proper(nil) -> ok(nil) 27.35/8.27 proper(half(X)) -> half(proper(X)) 27.35/8.27 terms(ok(X)) -> ok(terms(X)) 27.35/8.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 27.35/8.27 recip(ok(X)) -> ok(recip(X)) 27.35/8.27 sqr(ok(X)) -> ok(sqr(X)) 27.35/8.27 s(ok(X)) -> ok(s(X)) 27.35/8.27 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 27.35/8.27 dbl(ok(X)) -> ok(dbl(X)) 27.35/8.27 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 27.35/8.27 half(ok(X)) -> ok(half(X)) 27.35/8.27 top(mark(X)) -> top(proper(X)) 27.35/8.27 top(ok(X)) -> top(active(X)) 27.35/8.27 27.35/8.27 S is empty. 27.35/8.27 Rewrite Strategy: FULL 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (12) LowerBoundPropagationProof (FINISHED) 27.35/8.27 Propagated lower bound. 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (13) 27.35/8.27 BOUNDS(n^1, INF) 27.35/8.27 27.35/8.27 ---------------------------------------- 27.35/8.27 27.35/8.27 (14) 27.35/8.27 Obligation: 27.35/8.27 Analyzing the following TRS for decreasing loops: 27.35/8.27 27.35/8.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 27.35/8.27 27.35/8.27 27.35/8.27 The TRS R consists of the following rules: 27.35/8.27 27.35/8.27 active(terms(N)) -> mark(cons(recip(sqr(N)), terms(s(N)))) 27.35/8.27 active(sqr(0)) -> mark(0) 27.35/8.27 active(sqr(s(X))) -> mark(s(add(sqr(X), dbl(X)))) 27.35/8.27 active(dbl(0)) -> mark(0) 27.35/8.27 active(dbl(s(X))) -> mark(s(s(dbl(X)))) 27.35/8.27 active(add(0, X)) -> mark(X) 27.35/8.27 active(add(s(X), Y)) -> mark(s(add(X, Y))) 27.35/8.27 active(first(0, X)) -> mark(nil) 27.35/8.27 active(first(s(X), cons(Y, Z))) -> mark(cons(Y, first(X, Z))) 27.35/8.27 active(half(0)) -> mark(0) 27.35/8.27 active(half(s(0))) -> mark(0) 27.35/8.27 active(half(s(s(X)))) -> mark(s(half(X))) 27.35/8.27 active(half(dbl(X))) -> mark(X) 27.35/8.27 active(terms(X)) -> terms(active(X)) 27.35/8.27 active(cons(X1, X2)) -> cons(active(X1), X2) 27.35/8.27 active(recip(X)) -> recip(active(X)) 27.35/8.27 active(sqr(X)) -> sqr(active(X)) 27.35/8.27 active(s(X)) -> s(active(X)) 27.35/8.27 active(add(X1, X2)) -> add(active(X1), X2) 27.35/8.27 active(add(X1, X2)) -> add(X1, active(X2)) 27.35/8.27 active(dbl(X)) -> dbl(active(X)) 27.35/8.27 active(first(X1, X2)) -> first(active(X1), X2) 27.35/8.27 active(first(X1, X2)) -> first(X1, active(X2)) 27.35/8.27 active(half(X)) -> half(active(X)) 27.35/8.27 terms(mark(X)) -> mark(terms(X)) 27.35/8.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 27.35/8.27 recip(mark(X)) -> mark(recip(X)) 27.35/8.27 sqr(mark(X)) -> mark(sqr(X)) 27.35/8.27 s(mark(X)) -> mark(s(X)) 27.35/8.27 add(mark(X1), X2) -> mark(add(X1, X2)) 27.35/8.27 add(X1, mark(X2)) -> mark(add(X1, X2)) 27.35/8.27 dbl(mark(X)) -> mark(dbl(X)) 27.35/8.27 first(mark(X1), X2) -> mark(first(X1, X2)) 27.35/8.27 first(X1, mark(X2)) -> mark(first(X1, X2)) 27.35/8.27 half(mark(X)) -> mark(half(X)) 27.35/8.27 proper(terms(X)) -> terms(proper(X)) 27.35/8.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 27.35/8.27 proper(recip(X)) -> recip(proper(X)) 27.35/8.27 proper(sqr(X)) -> sqr(proper(X)) 27.35/8.27 proper(s(X)) -> s(proper(X)) 27.35/8.27 proper(0) -> ok(0) 27.35/8.27 proper(add(X1, X2)) -> add(proper(X1), proper(X2)) 27.35/8.27 proper(dbl(X)) -> dbl(proper(X)) 27.35/8.27 proper(first(X1, X2)) -> first(proper(X1), proper(X2)) 27.35/8.27 proper(nil) -> ok(nil) 27.35/8.27 proper(half(X)) -> half(proper(X)) 27.35/8.27 terms(ok(X)) -> ok(terms(X)) 27.35/8.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 27.35/8.27 recip(ok(X)) -> ok(recip(X)) 27.35/8.27 sqr(ok(X)) -> ok(sqr(X)) 27.35/8.27 s(ok(X)) -> ok(s(X)) 27.35/8.27 add(ok(X1), ok(X2)) -> ok(add(X1, X2)) 27.35/8.27 dbl(ok(X)) -> ok(dbl(X)) 27.35/8.27 first(ok(X1), ok(X2)) -> ok(first(X1, X2)) 27.35/8.27 half(ok(X)) -> ok(half(X)) 27.35/8.27 top(mark(X)) -> top(proper(X)) 27.35/8.27 top(ok(X)) -> top(active(X)) 27.35/8.27 27.35/8.27 S is empty. 27.35/8.27 Rewrite Strategy: FULL 27.50/8.31 EOF