1108.44/291.51 WORST_CASE(Omega(n^1), O(n^1)) 1123.57/295.27 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 1123.57/295.27 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1123.57/295.27 1123.57/295.27 1123.57/295.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1123.57/295.27 1123.57/295.27 (0) CpxTRS 1123.57/295.27 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms] 1123.57/295.27 (2) CpxTRS 1123.57/295.27 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 1123.57/295.27 (4) CpxTRS 1123.57/295.27 (5) CpxTrsMatchBoundsTAProof [FINISHED, 729 ms] 1123.57/295.27 (6) BOUNDS(1, n^1) 1123.57/295.27 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1123.57/295.27 (8) TRS for Loop Detection 1123.57/295.27 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1123.57/295.27 (10) BEST 1123.57/295.27 (11) proven lower bound 1123.57/295.27 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 1123.57/295.27 (13) BOUNDS(n^1, INF) 1123.57/295.27 (14) TRS for Loop Detection 1123.57/295.27 1123.57/295.27 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (0) 1123.57/295.27 Obligation: 1123.57/295.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1123.57/295.27 1123.57/295.27 1123.57/295.27 The TRS R consists of the following rules: 1123.57/295.27 1123.57/295.27 active(zeros) -> mark(cons(0, zeros)) 1123.57/295.27 active(and(tt, X)) -> mark(X) 1123.57/295.27 active(length(nil)) -> mark(0) 1123.57/295.27 active(length(cons(N, L))) -> mark(s(length(L))) 1123.57/295.27 active(take(0, IL)) -> mark(nil) 1123.57/295.27 active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL))) 1123.57/295.27 active(cons(X1, X2)) -> cons(active(X1), X2) 1123.57/295.27 active(and(X1, X2)) -> and(active(X1), X2) 1123.57/295.27 active(length(X)) -> length(active(X)) 1123.57/295.27 active(s(X)) -> s(active(X)) 1123.57/295.27 active(take(X1, X2)) -> take(active(X1), X2) 1123.57/295.27 active(take(X1, X2)) -> take(X1, active(X2)) 1123.57/295.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1123.57/295.27 and(mark(X1), X2) -> mark(and(X1, X2)) 1123.57/295.27 length(mark(X)) -> mark(length(X)) 1123.57/295.27 s(mark(X)) -> mark(s(X)) 1123.57/295.27 take(mark(X1), X2) -> mark(take(X1, X2)) 1123.57/295.27 take(X1, mark(X2)) -> mark(take(X1, X2)) 1123.57/295.27 proper(zeros) -> ok(zeros) 1123.57/295.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1123.57/295.27 proper(0) -> ok(0) 1123.57/295.27 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1123.57/295.27 proper(tt) -> ok(tt) 1123.57/295.27 proper(length(X)) -> length(proper(X)) 1123.57/295.27 proper(nil) -> ok(nil) 1123.57/295.27 proper(s(X)) -> s(proper(X)) 1123.57/295.27 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1123.57/295.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1123.57/295.27 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1123.57/295.27 length(ok(X)) -> ok(length(X)) 1123.57/295.27 s(ok(X)) -> ok(s(X)) 1123.57/295.27 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1123.57/295.27 top(mark(X)) -> top(proper(X)) 1123.57/295.27 top(ok(X)) -> top(active(X)) 1123.57/295.27 1123.57/295.27 S is empty. 1123.57/295.27 Rewrite Strategy: FULL 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 1123.57/295.27 The following defined symbols can occur below the 0th argument of cons: active, proper, cons 1123.57/295.27 The following defined symbols can occur below the 1th argument of cons: active, proper, cons 1123.57/295.27 The following defined symbols can occur below the 0th argument of top: active, proper, cons 1123.57/295.27 The following defined symbols can occur below the 0th argument of proper: active, proper, cons 1123.57/295.27 The following defined symbols can occur below the 0th argument of active: active, proper, cons 1123.57/295.27 1123.57/295.27 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 1123.57/295.27 active(and(tt, X)) -> mark(X) 1123.57/295.27 active(length(nil)) -> mark(0) 1123.57/295.27 active(length(cons(N, L))) -> mark(s(length(L))) 1123.57/295.27 active(take(0, IL)) -> mark(nil) 1123.57/295.27 active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL))) 1123.57/295.27 active(and(X1, X2)) -> and(active(X1), X2) 1123.57/295.27 active(length(X)) -> length(active(X)) 1123.57/295.27 active(s(X)) -> s(active(X)) 1123.57/295.27 active(take(X1, X2)) -> take(active(X1), X2) 1123.57/295.27 active(take(X1, X2)) -> take(X1, active(X2)) 1123.57/295.27 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1123.57/295.27 proper(length(X)) -> length(proper(X)) 1123.57/295.27 proper(s(X)) -> s(proper(X)) 1123.57/295.27 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1123.57/295.27 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (2) 1123.57/295.27 Obligation: 1123.57/295.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 1123.57/295.27 1123.57/295.27 1123.57/295.27 The TRS R consists of the following rules: 1123.57/295.27 1123.57/295.27 active(zeros) -> mark(cons(0, zeros)) 1123.57/295.27 active(cons(X1, X2)) -> cons(active(X1), X2) 1123.57/295.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1123.57/295.27 and(mark(X1), X2) -> mark(and(X1, X2)) 1123.57/295.27 length(mark(X)) -> mark(length(X)) 1123.57/295.27 s(mark(X)) -> mark(s(X)) 1123.57/295.27 take(mark(X1), X2) -> mark(take(X1, X2)) 1123.57/295.27 take(X1, mark(X2)) -> mark(take(X1, X2)) 1123.57/295.27 proper(zeros) -> ok(zeros) 1123.57/295.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1123.57/295.27 proper(0) -> ok(0) 1123.57/295.27 proper(tt) -> ok(tt) 1123.57/295.27 proper(nil) -> ok(nil) 1123.57/295.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1123.57/295.27 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1123.57/295.27 length(ok(X)) -> ok(length(X)) 1123.57/295.27 s(ok(X)) -> ok(s(X)) 1123.57/295.27 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1123.57/295.27 top(mark(X)) -> top(proper(X)) 1123.57/295.27 top(ok(X)) -> top(active(X)) 1123.57/295.27 1123.57/295.27 S is empty. 1123.57/295.27 Rewrite Strategy: FULL 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 1123.57/295.27 transformed relative TRS to TRS 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (4) 1123.57/295.27 Obligation: 1123.57/295.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 1123.57/295.27 1123.57/295.27 1123.57/295.27 The TRS R consists of the following rules: 1123.57/295.27 1123.57/295.27 active(zeros) -> mark(cons(0, zeros)) 1123.57/295.27 active(cons(X1, X2)) -> cons(active(X1), X2) 1123.57/295.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1123.57/295.27 and(mark(X1), X2) -> mark(and(X1, X2)) 1123.57/295.27 length(mark(X)) -> mark(length(X)) 1123.57/295.27 s(mark(X)) -> mark(s(X)) 1123.57/295.27 take(mark(X1), X2) -> mark(take(X1, X2)) 1123.57/295.27 take(X1, mark(X2)) -> mark(take(X1, X2)) 1123.57/295.27 proper(zeros) -> ok(zeros) 1123.57/295.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1123.57/295.27 proper(0) -> ok(0) 1123.57/295.27 proper(tt) -> ok(tt) 1123.57/295.27 proper(nil) -> ok(nil) 1123.57/295.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1123.57/295.27 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1123.57/295.27 length(ok(X)) -> ok(length(X)) 1123.57/295.27 s(ok(X)) -> ok(s(X)) 1123.57/295.27 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1123.57/295.27 top(mark(X)) -> top(proper(X)) 1123.57/295.27 top(ok(X)) -> top(active(X)) 1123.57/295.27 1123.57/295.27 S is empty. 1123.57/295.27 Rewrite Strategy: FULL 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (5) CpxTrsMatchBoundsTAProof (FINISHED) 1123.57/295.27 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 5. 1123.57/295.27 1123.57/295.27 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 1123.57/295.27 final states : [1, 2, 3, 4, 5, 6, 7, 8] 1123.57/295.27 transitions: 1123.57/295.27 zeros0() -> 0 1123.57/295.27 mark0(0) -> 0 1123.57/295.27 00() -> 0 1123.57/295.27 ok0(0) -> 0 1123.57/295.27 tt0() -> 0 1123.57/295.27 nil0() -> 0 1123.57/295.27 active0(0) -> 1 1123.57/295.27 cons0(0, 0) -> 2 1123.57/295.27 and0(0, 0) -> 3 1123.57/295.27 length0(0) -> 4 1123.57/295.27 s0(0) -> 5 1123.57/295.27 take0(0, 0) -> 6 1123.57/295.27 proper0(0) -> 7 1123.57/295.27 top0(0) -> 8 1123.57/295.27 01() -> 10 1123.57/295.27 zeros1() -> 11 1123.57/295.27 cons1(10, 11) -> 9 1123.57/295.27 mark1(9) -> 1 1123.57/295.27 cons1(0, 0) -> 12 1123.57/295.27 mark1(12) -> 2 1123.57/295.27 and1(0, 0) -> 13 1123.57/295.27 mark1(13) -> 3 1123.57/295.27 length1(0) -> 14 1123.57/295.27 mark1(14) -> 4 1123.57/295.27 s1(0) -> 15 1123.57/295.27 mark1(15) -> 5 1123.57/295.27 take1(0, 0) -> 16 1123.57/295.27 mark1(16) -> 6 1123.57/295.27 zeros1() -> 17 1123.57/295.27 ok1(17) -> 7 1123.57/295.27 01() -> 18 1123.57/295.27 ok1(18) -> 7 1123.57/295.27 tt1() -> 19 1123.57/295.27 ok1(19) -> 7 1123.57/295.27 nil1() -> 20 1123.57/295.27 ok1(20) -> 7 1123.57/295.27 cons1(0, 0) -> 21 1123.57/295.27 ok1(21) -> 2 1123.57/295.27 and1(0, 0) -> 22 1123.57/295.27 ok1(22) -> 3 1123.57/295.27 length1(0) -> 23 1123.57/295.27 ok1(23) -> 4 1123.57/295.27 s1(0) -> 24 1123.57/295.27 ok1(24) -> 5 1123.57/295.27 take1(0, 0) -> 25 1123.57/295.27 ok1(25) -> 6 1123.57/295.27 proper1(0) -> 26 1123.57/295.27 top1(26) -> 8 1123.57/295.27 active1(0) -> 27 1123.57/295.27 top1(27) -> 8 1123.57/295.27 mark1(9) -> 27 1123.57/295.27 mark1(12) -> 12 1123.57/295.27 mark1(12) -> 21 1123.57/295.27 mark1(13) -> 13 1123.57/295.27 mark1(13) -> 22 1123.57/295.27 mark1(14) -> 14 1123.57/295.27 mark1(14) -> 23 1123.57/295.27 mark1(15) -> 15 1123.57/295.27 mark1(15) -> 24 1123.57/295.27 mark1(16) -> 16 1123.57/295.27 mark1(16) -> 25 1123.57/295.27 ok1(17) -> 26 1123.57/295.27 ok1(18) -> 26 1123.57/295.27 ok1(19) -> 26 1123.57/295.27 ok1(20) -> 26 1123.57/295.27 ok1(21) -> 12 1123.57/295.27 ok1(21) -> 21 1123.57/295.27 ok1(22) -> 13 1123.57/295.27 ok1(22) -> 22 1123.57/295.27 ok1(23) -> 14 1123.57/295.27 ok1(23) -> 23 1123.57/295.27 ok1(24) -> 15 1123.57/295.27 ok1(24) -> 24 1123.57/295.27 ok1(25) -> 16 1123.57/295.27 ok1(25) -> 25 1123.57/295.27 proper2(9) -> 28 1123.57/295.27 top2(28) -> 8 1123.57/295.27 active2(17) -> 29 1123.57/295.27 top2(29) -> 8 1123.57/295.27 active2(18) -> 29 1123.57/295.27 active2(19) -> 29 1123.57/295.27 active2(20) -> 29 1123.57/295.27 02() -> 31 1123.57/295.27 zeros2() -> 32 1123.57/295.27 cons2(31, 32) -> 30 1123.57/295.27 mark2(30) -> 29 1123.57/295.27 proper2(10) -> 33 1123.57/295.27 proper2(11) -> 34 1123.57/295.27 cons2(33, 34) -> 28 1123.57/295.27 zeros2() -> 35 1123.57/295.27 ok2(35) -> 34 1123.57/295.27 02() -> 36 1123.57/295.27 ok2(36) -> 33 1123.57/295.27 proper3(30) -> 37 1123.57/295.27 top3(37) -> 8 1123.57/295.27 proper3(31) -> 38 1123.57/295.27 proper3(32) -> 39 1123.57/295.27 cons3(38, 39) -> 37 1123.57/295.27 cons3(36, 35) -> 40 1123.57/295.27 ok3(40) -> 28 1123.57/295.27 zeros3() -> 41 1123.57/295.27 ok3(41) -> 39 1123.57/295.27 03() -> 42 1123.57/295.27 ok3(42) -> 38 1123.57/295.27 active3(40) -> 43 1123.57/295.27 top3(43) -> 8 1123.57/295.27 cons4(42, 41) -> 44 1123.57/295.27 ok4(44) -> 37 1123.57/295.27 active4(36) -> 45 1123.57/295.27 cons4(45, 35) -> 43 1123.57/295.27 active4(44) -> 46 1123.57/295.27 top4(46) -> 8 1123.57/295.27 active5(42) -> 47 1123.57/295.27 cons5(47, 41) -> 46 1123.57/295.27 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (6) 1123.57/295.27 BOUNDS(1, n^1) 1123.57/295.27 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1123.57/295.27 Transformed a relative TRS into a decreasing-loop problem. 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (8) 1123.57/295.27 Obligation: 1123.57/295.27 Analyzing the following TRS for decreasing loops: 1123.57/295.27 1123.57/295.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1123.57/295.27 1123.57/295.27 1123.57/295.27 The TRS R consists of the following rules: 1123.57/295.27 1123.57/295.27 active(zeros) -> mark(cons(0, zeros)) 1123.57/295.27 active(and(tt, X)) -> mark(X) 1123.57/295.27 active(length(nil)) -> mark(0) 1123.57/295.27 active(length(cons(N, L))) -> mark(s(length(L))) 1123.57/295.27 active(take(0, IL)) -> mark(nil) 1123.57/295.27 active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL))) 1123.57/295.27 active(cons(X1, X2)) -> cons(active(X1), X2) 1123.57/295.27 active(and(X1, X2)) -> and(active(X1), X2) 1123.57/295.27 active(length(X)) -> length(active(X)) 1123.57/295.27 active(s(X)) -> s(active(X)) 1123.57/295.27 active(take(X1, X2)) -> take(active(X1), X2) 1123.57/295.27 active(take(X1, X2)) -> take(X1, active(X2)) 1123.57/295.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1123.57/295.27 and(mark(X1), X2) -> mark(and(X1, X2)) 1123.57/295.27 length(mark(X)) -> mark(length(X)) 1123.57/295.27 s(mark(X)) -> mark(s(X)) 1123.57/295.27 take(mark(X1), X2) -> mark(take(X1, X2)) 1123.57/295.27 take(X1, mark(X2)) -> mark(take(X1, X2)) 1123.57/295.27 proper(zeros) -> ok(zeros) 1123.57/295.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1123.57/295.27 proper(0) -> ok(0) 1123.57/295.27 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1123.57/295.27 proper(tt) -> ok(tt) 1123.57/295.27 proper(length(X)) -> length(proper(X)) 1123.57/295.27 proper(nil) -> ok(nil) 1123.57/295.27 proper(s(X)) -> s(proper(X)) 1123.57/295.27 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1123.57/295.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1123.57/295.27 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1123.57/295.27 length(ok(X)) -> ok(length(X)) 1123.57/295.27 s(ok(X)) -> ok(s(X)) 1123.57/295.27 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1123.57/295.27 top(mark(X)) -> top(proper(X)) 1123.57/295.27 top(ok(X)) -> top(active(X)) 1123.57/295.27 1123.57/295.27 S is empty. 1123.57/295.27 Rewrite Strategy: FULL 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (9) DecreasingLoopProof (LOWER BOUND(ID)) 1123.57/295.27 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1123.57/295.27 1123.57/295.27 The rewrite sequence 1123.57/295.27 1123.57/295.27 take(ok(X1), ok(X2)) ->^+ ok(take(X1, X2)) 1123.57/295.27 1123.57/295.27 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1123.57/295.27 1123.57/295.27 The pumping substitution is [X1 / ok(X1), X2 / ok(X2)]. 1123.57/295.27 1123.57/295.27 The result substitution is [ ]. 1123.57/295.27 1123.57/295.27 1123.57/295.27 1123.57/295.27 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (10) 1123.57/295.27 Complex Obligation (BEST) 1123.57/295.27 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (11) 1123.57/295.27 Obligation: 1123.57/295.27 Proved the lower bound n^1 for the following obligation: 1123.57/295.27 1123.57/295.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1123.57/295.27 1123.57/295.27 1123.57/295.27 The TRS R consists of the following rules: 1123.57/295.27 1123.57/295.27 active(zeros) -> mark(cons(0, zeros)) 1123.57/295.27 active(and(tt, X)) -> mark(X) 1123.57/295.27 active(length(nil)) -> mark(0) 1123.57/295.27 active(length(cons(N, L))) -> mark(s(length(L))) 1123.57/295.27 active(take(0, IL)) -> mark(nil) 1123.57/295.27 active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL))) 1123.57/295.27 active(cons(X1, X2)) -> cons(active(X1), X2) 1123.57/295.27 active(and(X1, X2)) -> and(active(X1), X2) 1123.57/295.27 active(length(X)) -> length(active(X)) 1123.57/295.27 active(s(X)) -> s(active(X)) 1123.57/295.27 active(take(X1, X2)) -> take(active(X1), X2) 1123.57/295.27 active(take(X1, X2)) -> take(X1, active(X2)) 1123.57/295.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1123.57/295.27 and(mark(X1), X2) -> mark(and(X1, X2)) 1123.57/295.27 length(mark(X)) -> mark(length(X)) 1123.57/295.27 s(mark(X)) -> mark(s(X)) 1123.57/295.27 take(mark(X1), X2) -> mark(take(X1, X2)) 1123.57/295.27 take(X1, mark(X2)) -> mark(take(X1, X2)) 1123.57/295.27 proper(zeros) -> ok(zeros) 1123.57/295.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1123.57/295.27 proper(0) -> ok(0) 1123.57/295.27 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1123.57/295.27 proper(tt) -> ok(tt) 1123.57/295.27 proper(length(X)) -> length(proper(X)) 1123.57/295.27 proper(nil) -> ok(nil) 1123.57/295.27 proper(s(X)) -> s(proper(X)) 1123.57/295.27 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1123.57/295.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1123.57/295.27 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1123.57/295.27 length(ok(X)) -> ok(length(X)) 1123.57/295.27 s(ok(X)) -> ok(s(X)) 1123.57/295.27 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1123.57/295.27 top(mark(X)) -> top(proper(X)) 1123.57/295.27 top(ok(X)) -> top(active(X)) 1123.57/295.27 1123.57/295.27 S is empty. 1123.57/295.27 Rewrite Strategy: FULL 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (12) LowerBoundPropagationProof (FINISHED) 1123.57/295.27 Propagated lower bound. 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (13) 1123.57/295.27 BOUNDS(n^1, INF) 1123.57/295.27 1123.57/295.27 ---------------------------------------- 1123.57/295.27 1123.57/295.27 (14) 1123.57/295.27 Obligation: 1123.57/295.27 Analyzing the following TRS for decreasing loops: 1123.57/295.27 1123.57/295.27 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1123.57/295.27 1123.57/295.27 1123.57/295.27 The TRS R consists of the following rules: 1123.57/295.27 1123.57/295.27 active(zeros) -> mark(cons(0, zeros)) 1123.57/295.27 active(and(tt, X)) -> mark(X) 1123.57/295.27 active(length(nil)) -> mark(0) 1123.57/295.27 active(length(cons(N, L))) -> mark(s(length(L))) 1123.57/295.27 active(take(0, IL)) -> mark(nil) 1123.57/295.27 active(take(s(M), cons(N, IL))) -> mark(cons(N, take(M, IL))) 1123.57/295.27 active(cons(X1, X2)) -> cons(active(X1), X2) 1123.57/295.27 active(and(X1, X2)) -> and(active(X1), X2) 1123.57/295.27 active(length(X)) -> length(active(X)) 1123.57/295.27 active(s(X)) -> s(active(X)) 1123.57/295.27 active(take(X1, X2)) -> take(active(X1), X2) 1123.57/295.27 active(take(X1, X2)) -> take(X1, active(X2)) 1123.57/295.27 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1123.57/295.27 and(mark(X1), X2) -> mark(and(X1, X2)) 1123.57/295.27 length(mark(X)) -> mark(length(X)) 1123.57/295.27 s(mark(X)) -> mark(s(X)) 1123.57/295.27 take(mark(X1), X2) -> mark(take(X1, X2)) 1123.57/295.27 take(X1, mark(X2)) -> mark(take(X1, X2)) 1123.57/295.27 proper(zeros) -> ok(zeros) 1123.57/295.27 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1123.57/295.27 proper(0) -> ok(0) 1123.57/295.27 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1123.57/295.27 proper(tt) -> ok(tt) 1123.57/295.27 proper(length(X)) -> length(proper(X)) 1123.57/295.27 proper(nil) -> ok(nil) 1123.57/295.27 proper(s(X)) -> s(proper(X)) 1123.57/295.27 proper(take(X1, X2)) -> take(proper(X1), proper(X2)) 1123.57/295.27 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1123.57/295.27 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1123.57/295.27 length(ok(X)) -> ok(length(X)) 1123.57/295.27 s(ok(X)) -> ok(s(X)) 1123.57/295.27 take(ok(X1), ok(X2)) -> ok(take(X1, X2)) 1123.57/295.27 top(mark(X)) -> top(proper(X)) 1123.57/295.27 top(ok(X)) -> top(active(X)) 1123.57/295.27 1123.57/295.27 S is empty. 1123.57/295.27 Rewrite Strategy: FULL 1123.57/295.33 EOF