23.23/7.16 WORST_CASE(Omega(n^1), O(n^1)) 23.23/7.17 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 23.23/7.17 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 23.23/7.17 23.23/7.17 23.23/7.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.23/7.17 23.23/7.17 (0) CpxTRS 23.23/7.17 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 8 ms] 23.23/7.17 (2) CpxTRS 23.23/7.17 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 23.23/7.17 (4) CpxTRS 23.23/7.17 (5) CpxTrsMatchBoundsTAProof [FINISHED, 34 ms] 23.23/7.17 (6) BOUNDS(1, n^1) 23.23/7.17 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 23.23/7.17 (8) TRS for Loop Detection 23.23/7.17 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 23.23/7.17 (10) BEST 23.23/7.17 (11) proven lower bound 23.23/7.17 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 23.23/7.17 (13) BOUNDS(n^1, INF) 23.23/7.17 (14) TRS for Loop Detection 23.23/7.17 23.23/7.17 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (0) 23.23/7.17 Obligation: 23.23/7.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.23/7.17 23.23/7.17 23.23/7.17 The TRS R consists of the following rules: 23.23/7.17 23.23/7.17 active(and(tt, X)) -> mark(X) 23.23/7.17 active(plus(N, 0)) -> mark(N) 23.23/7.17 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.23/7.17 active(and(X1, X2)) -> and(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.23/7.17 active(s(X)) -> s(active(X)) 23.23/7.17 and(mark(X1), X2) -> mark(and(X1, X2)) 23.23/7.17 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.23/7.17 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.23/7.17 s(mark(X)) -> mark(s(X)) 23.23/7.17 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.23/7.17 proper(tt) -> ok(tt) 23.23/7.17 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.23/7.17 proper(0) -> ok(0) 23.23/7.17 proper(s(X)) -> s(proper(X)) 23.23/7.17 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.23/7.17 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.23/7.17 s(ok(X)) -> ok(s(X)) 23.23/7.17 top(mark(X)) -> top(proper(X)) 23.23/7.17 top(ok(X)) -> top(active(X)) 23.23/7.17 23.23/7.17 S is empty. 23.23/7.17 Rewrite Strategy: FULL 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 23.23/7.17 The following defined symbols can occur below the 0th argument of top: proper, active 23.23/7.17 The following defined symbols can occur below the 0th argument of proper: proper, active 23.23/7.17 The following defined symbols can occur below the 0th argument of active: proper, active 23.23/7.17 23.23/7.17 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 23.23/7.17 active(and(tt, X)) -> mark(X) 23.23/7.17 active(plus(N, 0)) -> mark(N) 23.23/7.17 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.23/7.17 active(and(X1, X2)) -> and(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.23/7.17 active(s(X)) -> s(active(X)) 23.23/7.17 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.23/7.17 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.23/7.17 proper(s(X)) -> s(proper(X)) 23.23/7.17 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (2) 23.23/7.17 Obligation: 23.23/7.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.23/7.17 23.23/7.17 23.23/7.17 The TRS R consists of the following rules: 23.23/7.17 23.23/7.17 and(mark(X1), X2) -> mark(and(X1, X2)) 23.23/7.17 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.23/7.17 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.23/7.17 s(mark(X)) -> mark(s(X)) 23.23/7.17 proper(tt) -> ok(tt) 23.23/7.17 proper(0) -> ok(0) 23.23/7.17 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.23/7.17 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.23/7.17 s(ok(X)) -> ok(s(X)) 23.23/7.17 top(mark(X)) -> top(proper(X)) 23.23/7.17 top(ok(X)) -> top(active(X)) 23.23/7.17 23.23/7.17 S is empty. 23.23/7.17 Rewrite Strategy: FULL 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 23.23/7.17 transformed relative TRS to TRS 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (4) 23.23/7.17 Obligation: 23.23/7.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 23.23/7.17 23.23/7.17 23.23/7.17 The TRS R consists of the following rules: 23.23/7.17 23.23/7.17 and(mark(X1), X2) -> mark(and(X1, X2)) 23.23/7.17 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.23/7.17 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.23/7.17 s(mark(X)) -> mark(s(X)) 23.23/7.17 proper(tt) -> ok(tt) 23.23/7.17 proper(0) -> ok(0) 23.23/7.17 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.23/7.17 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.23/7.17 s(ok(X)) -> ok(s(X)) 23.23/7.17 top(mark(X)) -> top(proper(X)) 23.23/7.17 top(ok(X)) -> top(active(X)) 23.23/7.17 23.23/7.17 S is empty. 23.23/7.17 Rewrite Strategy: FULL 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (5) CpxTrsMatchBoundsTAProof (FINISHED) 23.23/7.17 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. 23.23/7.17 23.23/7.17 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 23.23/7.17 final states : [1, 2, 3, 4, 5] 23.23/7.17 transitions: 23.23/7.17 mark0(0) -> 0 23.23/7.17 tt0() -> 0 23.23/7.17 ok0(0) -> 0 23.23/7.17 00() -> 0 23.23/7.17 active0(0) -> 0 23.23/7.17 and0(0, 0) -> 1 23.23/7.17 plus0(0, 0) -> 2 23.23/7.17 s0(0) -> 3 23.23/7.17 proper0(0) -> 4 23.23/7.17 top0(0) -> 5 23.23/7.17 and1(0, 0) -> 6 23.23/7.17 mark1(6) -> 1 23.23/7.17 plus1(0, 0) -> 7 23.23/7.17 mark1(7) -> 2 23.23/7.17 s1(0) -> 8 23.23/7.17 mark1(8) -> 3 23.23/7.17 tt1() -> 9 23.23/7.17 ok1(9) -> 4 23.23/7.17 01() -> 10 23.23/7.17 ok1(10) -> 4 23.23/7.17 and1(0, 0) -> 11 23.23/7.17 ok1(11) -> 1 23.23/7.17 plus1(0, 0) -> 12 23.23/7.17 ok1(12) -> 2 23.23/7.17 s1(0) -> 13 23.23/7.17 ok1(13) -> 3 23.23/7.17 proper1(0) -> 14 23.23/7.17 top1(14) -> 5 23.23/7.17 active1(0) -> 15 23.23/7.17 top1(15) -> 5 23.23/7.17 mark1(6) -> 6 23.23/7.17 mark1(6) -> 11 23.23/7.17 mark1(7) -> 7 23.23/7.17 mark1(7) -> 12 23.23/7.17 mark1(8) -> 8 23.23/7.17 mark1(8) -> 13 23.23/7.17 ok1(9) -> 14 23.23/7.17 ok1(10) -> 14 23.23/7.17 ok1(11) -> 6 23.23/7.17 ok1(11) -> 11 23.23/7.17 ok1(12) -> 7 23.23/7.17 ok1(12) -> 12 23.23/7.17 ok1(13) -> 8 23.23/7.17 ok1(13) -> 13 23.23/7.17 active2(9) -> 16 23.23/7.17 top2(16) -> 5 23.23/7.17 active2(10) -> 16 23.23/7.17 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (6) 23.23/7.17 BOUNDS(1, n^1) 23.23/7.17 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 23.23/7.17 Transformed a relative TRS into a decreasing-loop problem. 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (8) 23.23/7.17 Obligation: 23.23/7.17 Analyzing the following TRS for decreasing loops: 23.23/7.17 23.23/7.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.23/7.17 23.23/7.17 23.23/7.17 The TRS R consists of the following rules: 23.23/7.17 23.23/7.17 active(and(tt, X)) -> mark(X) 23.23/7.17 active(plus(N, 0)) -> mark(N) 23.23/7.17 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.23/7.17 active(and(X1, X2)) -> and(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.23/7.17 active(s(X)) -> s(active(X)) 23.23/7.17 and(mark(X1), X2) -> mark(and(X1, X2)) 23.23/7.17 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.23/7.17 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.23/7.17 s(mark(X)) -> mark(s(X)) 23.23/7.17 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.23/7.17 proper(tt) -> ok(tt) 23.23/7.17 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.23/7.17 proper(0) -> ok(0) 23.23/7.17 proper(s(X)) -> s(proper(X)) 23.23/7.17 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.23/7.17 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.23/7.17 s(ok(X)) -> ok(s(X)) 23.23/7.17 top(mark(X)) -> top(proper(X)) 23.23/7.17 top(ok(X)) -> top(active(X)) 23.23/7.17 23.23/7.17 S is empty. 23.23/7.17 Rewrite Strategy: FULL 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (9) DecreasingLoopProof (LOWER BOUND(ID)) 23.23/7.17 The following loop(s) give(s) rise to the lower bound Omega(n^1): 23.23/7.17 23.23/7.17 The rewrite sequence 23.23/7.17 23.23/7.17 plus(X1, mark(X2)) ->^+ mark(plus(X1, X2)) 23.23/7.17 23.23/7.17 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 23.23/7.17 23.23/7.17 The pumping substitution is [X2 / mark(X2)]. 23.23/7.17 23.23/7.17 The result substitution is [ ]. 23.23/7.17 23.23/7.17 23.23/7.17 23.23/7.17 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (10) 23.23/7.17 Complex Obligation (BEST) 23.23/7.17 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (11) 23.23/7.17 Obligation: 23.23/7.17 Proved the lower bound n^1 for the following obligation: 23.23/7.17 23.23/7.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.23/7.17 23.23/7.17 23.23/7.17 The TRS R consists of the following rules: 23.23/7.17 23.23/7.17 active(and(tt, X)) -> mark(X) 23.23/7.17 active(plus(N, 0)) -> mark(N) 23.23/7.17 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.23/7.17 active(and(X1, X2)) -> and(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.23/7.17 active(s(X)) -> s(active(X)) 23.23/7.17 and(mark(X1), X2) -> mark(and(X1, X2)) 23.23/7.17 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.23/7.17 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.23/7.17 s(mark(X)) -> mark(s(X)) 23.23/7.17 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.23/7.17 proper(tt) -> ok(tt) 23.23/7.17 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.23/7.17 proper(0) -> ok(0) 23.23/7.17 proper(s(X)) -> s(proper(X)) 23.23/7.17 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.23/7.17 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.23/7.17 s(ok(X)) -> ok(s(X)) 23.23/7.17 top(mark(X)) -> top(proper(X)) 23.23/7.17 top(ok(X)) -> top(active(X)) 23.23/7.17 23.23/7.17 S is empty. 23.23/7.17 Rewrite Strategy: FULL 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (12) LowerBoundPropagationProof (FINISHED) 23.23/7.17 Propagated lower bound. 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (13) 23.23/7.17 BOUNDS(n^1, INF) 23.23/7.17 23.23/7.17 ---------------------------------------- 23.23/7.17 23.23/7.17 (14) 23.23/7.17 Obligation: 23.23/7.17 Analyzing the following TRS for decreasing loops: 23.23/7.17 23.23/7.17 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 23.23/7.17 23.23/7.17 23.23/7.17 The TRS R consists of the following rules: 23.23/7.17 23.23/7.17 active(and(tt, X)) -> mark(X) 23.23/7.17 active(plus(N, 0)) -> mark(N) 23.23/7.17 active(plus(N, s(M))) -> mark(s(plus(N, M))) 23.23/7.17 active(and(X1, X2)) -> and(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(active(X1), X2) 23.23/7.17 active(plus(X1, X2)) -> plus(X1, active(X2)) 23.23/7.17 active(s(X)) -> s(active(X)) 23.23/7.17 and(mark(X1), X2) -> mark(and(X1, X2)) 23.23/7.17 plus(mark(X1), X2) -> mark(plus(X1, X2)) 23.23/7.17 plus(X1, mark(X2)) -> mark(plus(X1, X2)) 23.23/7.17 s(mark(X)) -> mark(s(X)) 23.23/7.17 proper(and(X1, X2)) -> and(proper(X1), proper(X2)) 23.23/7.17 proper(tt) -> ok(tt) 23.23/7.17 proper(plus(X1, X2)) -> plus(proper(X1), proper(X2)) 23.23/7.17 proper(0) -> ok(0) 23.23/7.17 proper(s(X)) -> s(proper(X)) 23.23/7.17 and(ok(X1), ok(X2)) -> ok(and(X1, X2)) 23.23/7.17 plus(ok(X1), ok(X2)) -> ok(plus(X1, X2)) 23.23/7.17 s(ok(X)) -> ok(s(X)) 23.23/7.17 top(mark(X)) -> top(proper(X)) 23.23/7.17 top(ok(X)) -> top(active(X)) 23.23/7.17 23.23/7.17 S is empty. 23.23/7.17 Rewrite Strategy: FULL 24.99/10.31 EOF