1102.02/291.52 WORST_CASE(Omega(n^1), O(n^1)) 1102.88/291.79 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 1102.88/291.79 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 1102.88/291.79 1102.88/291.79 1102.88/291.79 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1102.88/291.79 1102.88/291.79 (0) CpxTRS 1102.88/291.79 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 11 ms] 1102.88/291.79 (2) CpxTRS 1102.88/291.79 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 1102.88/291.79 (4) CpxTRS 1102.88/291.79 (5) CpxTrsMatchBoundsTAProof [FINISHED, 512 ms] 1102.88/291.79 (6) BOUNDS(1, n^1) 1102.88/291.79 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 1102.88/291.79 (8) TRS for Loop Detection 1102.88/291.79 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 1102.88/291.79 (10) BEST 1102.88/291.79 (11) proven lower bound 1102.88/291.79 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 1102.88/291.79 (13) BOUNDS(n^1, INF) 1102.88/291.79 (14) TRS for Loop Detection 1102.88/291.79 1102.88/291.79 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (0) 1102.88/291.79 Obligation: 1102.88/291.79 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1102.88/291.79 1102.88/291.79 1102.88/291.79 The TRS R consists of the following rules: 1102.88/291.79 1102.88/291.79 active(zeros) -> mark(cons(0, zeros)) 1102.88/291.79 active(and(tt, X)) -> mark(X) 1102.88/291.79 active(length(nil)) -> mark(0) 1102.88/291.79 active(length(cons(N, L))) -> mark(s(length(L))) 1102.88/291.79 active(cons(X1, X2)) -> cons(active(X1), X2) 1102.88/291.79 active(and(X1, X2)) -> and(active(X1), X2) 1102.88/291.79 active(length(X)) -> length(active(X)) 1102.88/291.79 active(s(X)) -> s(active(X)) 1102.88/291.79 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1102.88/291.79 and(mark(X1), X2) -> mark(and(X1, X2)) 1102.88/291.79 length(mark(X)) -> mark(length(X)) 1102.88/291.79 s(mark(X)) -> mark(s(X)) 1102.88/291.79 proper(zeros) -> ok(zeros) 1102.88/291.79 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1102.88/291.79 proper(0) -> ok(0) 1102.88/291.79 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1102.88/291.79 proper(tt) -> ok(tt) 1102.88/291.79 proper(length(X)) -> length(proper(X)) 1102.88/291.79 proper(nil) -> ok(nil) 1102.88/291.79 proper(s(X)) -> s(proper(X)) 1102.88/291.79 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1102.88/291.79 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1102.88/291.79 length(ok(X)) -> ok(length(X)) 1102.88/291.79 s(ok(X)) -> ok(s(X)) 1102.88/291.79 top(mark(X)) -> top(proper(X)) 1102.88/291.79 top(ok(X)) -> top(active(X)) 1102.88/291.79 1102.88/291.79 S is empty. 1102.88/291.79 Rewrite Strategy: FULL 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 1102.88/291.79 The following defined symbols can occur below the 0th argument of cons: active, proper, cons 1102.88/291.79 The following defined symbols can occur below the 1th argument of cons: active, proper, cons 1102.88/291.79 The following defined symbols can occur below the 0th argument of top: active, proper, cons 1102.88/291.79 The following defined symbols can occur below the 0th argument of proper: active, proper, cons 1102.88/291.79 The following defined symbols can occur below the 0th argument of active: active, proper, cons 1102.88/291.79 1102.88/291.79 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 1102.88/291.79 active(and(tt, X)) -> mark(X) 1102.88/291.79 active(length(nil)) -> mark(0) 1102.88/291.79 active(length(cons(N, L))) -> mark(s(length(L))) 1102.88/291.79 active(and(X1, X2)) -> and(active(X1), X2) 1102.88/291.79 active(length(X)) -> length(active(X)) 1102.88/291.79 active(s(X)) -> s(active(X)) 1102.88/291.79 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1102.88/291.79 proper(length(X)) -> length(proper(X)) 1102.88/291.79 proper(s(X)) -> s(proper(X)) 1102.88/291.79 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (2) 1102.88/291.79 Obligation: 1102.88/291.79 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 1102.88/291.79 1102.88/291.79 1102.88/291.79 The TRS R consists of the following rules: 1102.88/291.79 1102.88/291.79 active(zeros) -> mark(cons(0, zeros)) 1102.88/291.79 active(cons(X1, X2)) -> cons(active(X1), X2) 1102.88/291.79 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1102.88/291.79 and(mark(X1), X2) -> mark(and(X1, X2)) 1102.88/291.79 length(mark(X)) -> mark(length(X)) 1102.88/291.79 s(mark(X)) -> mark(s(X)) 1102.88/291.79 proper(zeros) -> ok(zeros) 1102.88/291.79 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1102.88/291.79 proper(0) -> ok(0) 1102.88/291.79 proper(tt) -> ok(tt) 1102.88/291.79 proper(nil) -> ok(nil) 1102.88/291.79 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1102.88/291.79 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1102.88/291.79 length(ok(X)) -> ok(length(X)) 1102.88/291.79 s(ok(X)) -> ok(s(X)) 1102.88/291.79 top(mark(X)) -> top(proper(X)) 1102.88/291.79 top(ok(X)) -> top(active(X)) 1102.88/291.79 1102.88/291.79 S is empty. 1102.88/291.79 Rewrite Strategy: FULL 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 1102.88/291.79 transformed relative TRS to TRS 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (4) 1102.88/291.79 Obligation: 1102.88/291.79 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 1102.88/291.79 1102.88/291.79 1102.88/291.79 The TRS R consists of the following rules: 1102.88/291.79 1102.88/291.79 active(zeros) -> mark(cons(0, zeros)) 1102.88/291.79 active(cons(X1, X2)) -> cons(active(X1), X2) 1102.88/291.79 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1102.88/291.79 and(mark(X1), X2) -> mark(and(X1, X2)) 1102.88/291.79 length(mark(X)) -> mark(length(X)) 1102.88/291.79 s(mark(X)) -> mark(s(X)) 1102.88/291.79 proper(zeros) -> ok(zeros) 1102.88/291.79 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1102.88/291.79 proper(0) -> ok(0) 1102.88/291.79 proper(tt) -> ok(tt) 1102.88/291.79 proper(nil) -> ok(nil) 1102.88/291.79 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1102.88/291.79 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1102.88/291.79 length(ok(X)) -> ok(length(X)) 1102.88/291.79 s(ok(X)) -> ok(s(X)) 1102.88/291.79 top(mark(X)) -> top(proper(X)) 1102.88/291.79 top(ok(X)) -> top(active(X)) 1102.88/291.79 1102.88/291.79 S is empty. 1102.88/291.79 Rewrite Strategy: FULL 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (5) CpxTrsMatchBoundsTAProof (FINISHED) 1102.88/291.79 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. 1102.88/291.79 1102.88/291.79 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 1102.88/291.79 final states : [1, 2, 3, 4, 5, 6, 7] 1102.88/291.79 transitions: 1102.88/291.79 zeros0() -> 0 1102.88/291.79 mark0(0) -> 0 1102.88/291.79 00() -> 0 1102.88/291.79 ok0(0) -> 0 1102.88/291.79 tt0() -> 0 1102.88/291.79 nil0() -> 0 1102.88/291.79 active0(0) -> 1 1102.88/291.79 cons0(0, 0) -> 2 1102.88/291.79 and0(0, 0) -> 3 1102.88/291.79 length0(0) -> 4 1102.88/291.79 s0(0) -> 5 1102.88/291.79 proper0(0) -> 6 1102.88/291.79 top0(0) -> 7 1102.88/291.79 01() -> 9 1102.88/291.79 zeros1() -> 10 1102.88/291.79 cons1(9, 10) -> 8 1102.88/291.79 mark1(8) -> 1 1102.88/291.79 cons1(0, 0) -> 11 1102.88/291.79 mark1(11) -> 2 1102.88/291.79 and1(0, 0) -> 12 1102.88/291.79 mark1(12) -> 3 1102.88/291.79 length1(0) -> 13 1102.88/291.79 mark1(13) -> 4 1102.88/291.79 s1(0) -> 14 1102.88/291.79 mark1(14) -> 5 1102.88/291.79 zeros1() -> 15 1102.88/291.79 ok1(15) -> 6 1102.88/291.79 01() -> 16 1102.88/291.79 ok1(16) -> 6 1102.88/291.79 tt1() -> 17 1102.88/291.79 ok1(17) -> 6 1102.88/291.79 nil1() -> 18 1102.88/291.79 ok1(18) -> 6 1102.88/291.79 cons1(0, 0) -> 19 1102.88/291.79 ok1(19) -> 2 1102.88/291.79 and1(0, 0) -> 20 1102.88/291.79 ok1(20) -> 3 1102.88/291.79 length1(0) -> 21 1102.88/291.79 ok1(21) -> 4 1102.88/291.79 s1(0) -> 22 1102.88/291.79 ok1(22) -> 5 1102.88/291.79 proper1(0) -> 23 1102.88/291.79 top1(23) -> 7 1102.88/291.79 active1(0) -> 24 1102.88/291.79 top1(24) -> 7 1102.88/291.79 mark1(8) -> 24 1102.88/291.79 mark1(11) -> 11 1102.88/291.79 mark1(11) -> 19 1102.88/291.79 mark1(12) -> 12 1102.88/291.79 mark1(12) -> 20 1102.88/291.79 mark1(13) -> 13 1102.88/291.79 mark1(13) -> 21 1102.88/291.79 mark1(14) -> 14 1102.88/291.79 mark1(14) -> 22 1102.88/291.79 ok1(15) -> 23 1102.88/291.79 ok1(16) -> 23 1102.88/291.79 ok1(17) -> 23 1102.88/291.79 ok1(18) -> 23 1102.88/291.79 ok1(19) -> 11 1102.88/291.79 ok1(19) -> 19 1102.88/291.79 ok1(20) -> 12 1102.88/291.79 ok1(20) -> 20 1102.88/291.79 ok1(21) -> 13 1102.88/291.79 ok1(21) -> 21 1102.88/291.79 ok1(22) -> 14 1102.88/291.79 ok1(22) -> 22 1102.88/291.79 proper2(8) -> 25 1102.88/291.79 top2(25) -> 7 1102.88/291.79 active2(15) -> 26 1102.88/291.79 top2(26) -> 7 1102.88/291.79 active2(16) -> 26 1102.88/291.79 active2(17) -> 26 1102.88/291.79 active2(18) -> 26 1102.88/291.79 02() -> 28 1102.88/291.79 zeros2() -> 29 1102.88/291.79 cons2(28, 29) -> 27 1102.88/291.79 mark2(27) -> 26 1102.88/291.79 proper2(9) -> 30 1102.88/291.79 proper2(10) -> 31 1102.88/291.79 cons2(30, 31) -> 25 1102.88/291.79 zeros2() -> 32 1102.88/291.79 ok2(32) -> 31 1102.88/291.79 02() -> 33 1102.88/291.79 ok2(33) -> 30 1102.88/291.79 proper3(27) -> 34 1102.88/291.79 top3(34) -> 7 1102.88/291.79 proper3(28) -> 35 1102.88/291.79 proper3(29) -> 36 1102.88/291.79 cons3(35, 36) -> 34 1102.88/291.79 cons3(33, 32) -> 37 1102.88/291.79 ok3(37) -> 25 1102.88/291.79 zeros3() -> 38 1102.88/291.79 ok3(38) -> 36 1102.88/291.79 03() -> 39 1102.88/291.79 ok3(39) -> 35 1102.88/291.79 active3(37) -> 40 1102.88/291.79 top3(40) -> 7 1102.88/291.79 cons4(39, 38) -> 41 1102.88/291.79 ok4(41) -> 34 1102.88/291.79 active4(33) -> 42 1102.88/291.79 cons4(42, 32) -> 40 1102.88/291.79 active4(41) -> 43 1102.88/291.79 top4(43) -> 7 1102.88/291.79 active5(39) -> 44 1102.88/291.79 cons5(44, 38) -> 43 1102.88/291.79 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (6) 1102.88/291.79 BOUNDS(1, n^1) 1102.88/291.79 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 1102.88/291.79 Transformed a relative TRS into a decreasing-loop problem. 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (8) 1102.88/291.79 Obligation: 1102.88/291.79 Analyzing the following TRS for decreasing loops: 1102.88/291.79 1102.88/291.79 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1102.88/291.79 1102.88/291.79 1102.88/291.79 The TRS R consists of the following rules: 1102.88/291.79 1102.88/291.79 active(zeros) -> mark(cons(0, zeros)) 1102.88/291.79 active(and(tt, X)) -> mark(X) 1102.88/291.79 active(length(nil)) -> mark(0) 1102.88/291.79 active(length(cons(N, L))) -> mark(s(length(L))) 1102.88/291.79 active(cons(X1, X2)) -> cons(active(X1), X2) 1102.88/291.79 active(and(X1, X2)) -> and(active(X1), X2) 1102.88/291.79 active(length(X)) -> length(active(X)) 1102.88/291.79 active(s(X)) -> s(active(X)) 1102.88/291.79 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1102.88/291.79 and(mark(X1), X2) -> mark(and(X1, X2)) 1102.88/291.79 length(mark(X)) -> mark(length(X)) 1102.88/291.79 s(mark(X)) -> mark(s(X)) 1102.88/291.79 proper(zeros) -> ok(zeros) 1102.88/291.79 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1102.88/291.79 proper(0) -> ok(0) 1102.88/291.79 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1102.88/291.79 proper(tt) -> ok(tt) 1102.88/291.79 proper(length(X)) -> length(proper(X)) 1102.88/291.79 proper(nil) -> ok(nil) 1102.88/291.79 proper(s(X)) -> s(proper(X)) 1102.88/291.79 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1102.88/291.79 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1102.88/291.79 length(ok(X)) -> ok(length(X)) 1102.88/291.79 s(ok(X)) -> ok(s(X)) 1102.88/291.79 top(mark(X)) -> top(proper(X)) 1102.88/291.79 top(ok(X)) -> top(active(X)) 1102.88/291.79 1102.88/291.79 S is empty. 1102.88/291.79 Rewrite Strategy: FULL 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (9) DecreasingLoopProof (LOWER BOUND(ID)) 1102.88/291.79 The following loop(s) give(s) rise to the lower bound Omega(n^1): 1102.88/291.79 1102.88/291.79 The rewrite sequence 1102.88/291.79 1102.88/291.79 s(mark(X)) ->^+ mark(s(X)) 1102.88/291.79 1102.88/291.79 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 1102.88/291.79 1102.88/291.79 The pumping substitution is [X / mark(X)]. 1102.88/291.79 1102.88/291.79 The result substitution is [ ]. 1102.88/291.79 1102.88/291.79 1102.88/291.79 1102.88/291.79 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (10) 1102.88/291.79 Complex Obligation (BEST) 1102.88/291.79 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (11) 1102.88/291.79 Obligation: 1102.88/291.79 Proved the lower bound n^1 for the following obligation: 1102.88/291.79 1102.88/291.79 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1102.88/291.79 1102.88/291.79 1102.88/291.79 The TRS R consists of the following rules: 1102.88/291.79 1102.88/291.79 active(zeros) -> mark(cons(0, zeros)) 1102.88/291.79 active(and(tt, X)) -> mark(X) 1102.88/291.79 active(length(nil)) -> mark(0) 1102.88/291.79 active(length(cons(N, L))) -> mark(s(length(L))) 1102.88/291.79 active(cons(X1, X2)) -> cons(active(X1), X2) 1102.88/291.79 active(and(X1, X2)) -> and(active(X1), X2) 1102.88/291.79 active(length(X)) -> length(active(X)) 1102.88/291.79 active(s(X)) -> s(active(X)) 1102.88/291.79 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1102.88/291.79 and(mark(X1), X2) -> mark(and(X1, X2)) 1102.88/291.79 length(mark(X)) -> mark(length(X)) 1102.88/291.79 s(mark(X)) -> mark(s(X)) 1102.88/291.79 proper(zeros) -> ok(zeros) 1102.88/291.79 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1102.88/291.79 proper(0) -> ok(0) 1102.88/291.79 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1102.88/291.79 proper(tt) -> ok(tt) 1102.88/291.79 proper(length(X)) -> length(proper(X)) 1102.88/291.79 proper(nil) -> ok(nil) 1102.88/291.79 proper(s(X)) -> s(proper(X)) 1102.88/291.79 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1102.88/291.79 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1102.88/291.79 length(ok(X)) -> ok(length(X)) 1102.88/291.79 s(ok(X)) -> ok(s(X)) 1102.88/291.79 top(mark(X)) -> top(proper(X)) 1102.88/291.79 top(ok(X)) -> top(active(X)) 1102.88/291.79 1102.88/291.79 S is empty. 1102.88/291.79 Rewrite Strategy: FULL 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (12) LowerBoundPropagationProof (FINISHED) 1102.88/291.79 Propagated lower bound. 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (13) 1102.88/291.79 BOUNDS(n^1, INF) 1102.88/291.79 1102.88/291.79 ---------------------------------------- 1102.88/291.79 1102.88/291.79 (14) 1102.88/291.79 Obligation: 1102.88/291.79 Analyzing the following TRS for decreasing loops: 1102.88/291.79 1102.88/291.79 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 1102.88/291.79 1102.88/291.79 1102.88/291.79 The TRS R consists of the following rules: 1102.88/291.79 1102.88/291.79 active(zeros) -> mark(cons(0, zeros)) 1102.88/291.79 active(and(tt, X)) -> mark(X) 1102.88/291.79 active(length(nil)) -> mark(0) 1102.88/291.79 active(length(cons(N, L))) -> mark(s(length(L))) 1102.88/291.79 active(cons(X1, X2)) -> cons(active(X1), X2) 1102.88/291.79 active(and(X1, X2)) -> and(active(X1), X2) 1102.88/291.79 active(length(X)) -> length(active(X)) 1102.88/291.79 active(s(X)) -> s(active(X)) 1102.88/291.79 cons(mark(X1), X2) -> mark(cons(X1, X2)) 1102.88/291.79 and(mark(X1), X2) -> mark(and(X1, X2)) 1102.88/291.79 length(mark(X)) -> mark(length(X)) 1102.88/291.79 s(mark(X)) -> mark(s(X)) 1102.88/291.79 proper(zeros) -> ok(zeros) 1102.88/291.79 proper(cons(X1, X2)) -> cons(proper(X1), proper(X2)) 1102.88/291.79 proper(0) -> ok(0) 1102.88/291.79 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 1102.88/291.79 proper(tt) -> ok(tt) 1102.88/291.79 proper(length(X)) -> length(proper(X)) 1102.88/291.79 proper(nil) -> ok(nil) 1102.88/291.79 proper(s(X)) -> s(proper(X)) 1102.88/291.79 cons(ok(X1), ok(X2)) -> ok(cons(X1, X2)) 1102.88/291.79 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 1102.88/291.79 length(ok(X)) -> ok(length(X)) 1102.88/291.79 s(ok(X)) -> ok(s(X)) 1102.88/291.79 top(mark(X)) -> top(proper(X)) 1102.88/291.79 top(ok(X)) -> top(active(X)) 1102.88/291.79 1102.88/291.79 S is empty. 1102.88/291.79 Rewrite Strategy: FULL 1103.16/291.86 EOF