25.58/8.94 WORST_CASE(Omega(n^1), O(n^1)) 25.58/8.95 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 25.58/8.95 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 25.58/8.95 25.58/8.95 25.58/8.95 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.58/8.95 25.58/8.95 (0) CpxTRS 25.58/8.95 (1) NestedDefinedSymbolProof [UPPER BOUND(ID), 1 ms] 25.58/8.95 (2) CpxTRS 25.58/8.95 (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] 25.58/8.95 (4) CpxTRS 25.58/8.95 (5) CpxTrsMatchBoundsTAProof [FINISHED, 185 ms] 25.58/8.95 (6) BOUNDS(1, n^1) 25.58/8.95 (7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 25.58/8.95 (8) TRS for Loop Detection 25.58/8.95 (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 25.58/8.95 (10) BEST 25.58/8.95 (11) proven lower bound 25.58/8.95 (12) LowerBoundPropagationProof [FINISHED, 0 ms] 25.58/8.95 (13) BOUNDS(n^1, INF) 25.58/8.95 (14) TRS for Loop Detection 25.58/8.95 25.58/8.95 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (0) 25.58/8.95 Obligation: 25.58/8.95 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.58/8.95 25.58/8.95 25.58/8.95 The TRS R consists of the following rules: 25.58/8.95 25.58/8.95 active(h(X)) -> mark(g(X, X)) 25.58/8.95 active(g(a, X)) -> mark(f(b, X)) 25.58/8.95 active(f(X, X)) -> mark(h(a)) 25.58/8.95 active(a) -> mark(b) 25.58/8.95 active(h(X)) -> h(active(X)) 25.58/8.95 active(g(X1, X2)) -> g(active(X1), X2) 25.58/8.95 active(f(X1, X2)) -> f(active(X1), X2) 25.58/8.95 h(mark(X)) -> mark(h(X)) 25.58/8.95 g(mark(X1), X2) -> mark(g(X1, X2)) 25.58/8.95 f(mark(X1), X2) -> mark(f(X1, X2)) 25.58/8.95 proper(h(X)) -> h(proper(X)) 25.58/8.95 proper(g(X1, X2)) -> g(proper(X1), proper(X2)) 25.58/8.95 proper(a) -> ok(a) 25.58/8.95 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 25.58/8.95 proper(b) -> ok(b) 25.58/8.95 h(ok(X)) -> ok(h(X)) 25.58/8.95 g(ok(X1), ok(X2)) -> ok(g(X1, X2)) 25.58/8.95 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 25.58/8.95 top(mark(X)) -> top(proper(X)) 25.58/8.95 top(ok(X)) -> top(active(X)) 25.58/8.95 25.58/8.95 S is empty. 25.58/8.95 Rewrite Strategy: FULL 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (1) NestedDefinedSymbolProof (UPPER BOUND(ID)) 25.58/8.95 The following defined symbols can occur below the 0th argument of top: proper, active 25.58/8.95 The following defined symbols can occur below the 0th argument of proper: proper, active 25.58/8.95 The following defined symbols can occur below the 0th argument of active: proper, active 25.58/8.95 25.58/8.95 Hence, the left-hand sides of the following rules are not basic-reachable and can be removed: 25.58/8.95 active(h(X)) -> mark(g(X, X)) 25.58/8.95 active(g(a, X)) -> mark(f(b, X)) 25.58/8.95 active(f(X, X)) -> mark(h(a)) 25.58/8.95 active(h(X)) -> h(active(X)) 25.58/8.95 active(g(X1, X2)) -> g(active(X1), X2) 25.58/8.95 active(f(X1, X2)) -> f(active(X1), X2) 25.58/8.95 proper(h(X)) -> h(proper(X)) 25.58/8.95 proper(g(X1, X2)) -> g(proper(X1), proper(X2)) 25.58/8.95 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 25.58/8.95 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (2) 25.58/8.95 Obligation: 25.58/8.95 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 25.58/8.95 25.58/8.95 25.58/8.95 The TRS R consists of the following rules: 25.58/8.95 25.58/8.95 active(a) -> mark(b) 25.58/8.95 h(mark(X)) -> mark(h(X)) 25.58/8.95 g(mark(X1), X2) -> mark(g(X1, X2)) 25.58/8.95 f(mark(X1), X2) -> mark(f(X1, X2)) 25.58/8.95 proper(a) -> ok(a) 25.58/8.95 proper(b) -> ok(b) 25.58/8.95 h(ok(X)) -> ok(h(X)) 25.58/8.95 g(ok(X1), ok(X2)) -> ok(g(X1, X2)) 25.58/8.95 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 25.58/8.95 top(mark(X)) -> top(proper(X)) 25.58/8.95 top(ok(X)) -> top(active(X)) 25.58/8.95 25.58/8.95 S is empty. 25.58/8.95 Rewrite Strategy: FULL 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (3) RelTrsToTrsProof (UPPER BOUND(ID)) 25.58/8.95 transformed relative TRS to TRS 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (4) 25.58/8.95 Obligation: 25.58/8.95 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). 25.58/8.95 25.58/8.95 25.58/8.95 The TRS R consists of the following rules: 25.58/8.95 25.58/8.95 active(a) -> mark(b) 25.58/8.95 h(mark(X)) -> mark(h(X)) 25.58/8.95 g(mark(X1), X2) -> mark(g(X1, X2)) 25.58/8.95 f(mark(X1), X2) -> mark(f(X1, X2)) 25.58/8.95 proper(a) -> ok(a) 25.58/8.95 proper(b) -> ok(b) 25.58/8.95 h(ok(X)) -> ok(h(X)) 25.58/8.95 g(ok(X1), ok(X2)) -> ok(g(X1, X2)) 25.58/8.95 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 25.58/8.95 top(mark(X)) -> top(proper(X)) 25.58/8.95 top(ok(X)) -> top(active(X)) 25.58/8.95 25.58/8.95 S is empty. 25.58/8.95 Rewrite Strategy: FULL 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (5) CpxTrsMatchBoundsTAProof (FINISHED) 25.58/8.95 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 4. 25.58/8.95 25.58/8.95 The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 25.58/8.95 final states : [1, 2, 3, 4, 5, 6] 25.58/8.95 transitions: 25.58/8.95 a0() -> 0 25.58/8.95 mark0(0) -> 0 25.58/8.95 b0() -> 0 25.58/8.95 ok0(0) -> 0 25.58/8.95 active0(0) -> 1 25.58/8.95 h0(0) -> 2 25.58/8.95 g0(0, 0) -> 3 25.58/8.95 f0(0, 0) -> 4 25.58/8.95 proper0(0) -> 5 25.58/8.95 top0(0) -> 6 25.58/8.95 b1() -> 7 25.58/8.95 mark1(7) -> 1 25.58/8.95 h1(0) -> 8 25.58/8.95 mark1(8) -> 2 25.58/8.95 g1(0, 0) -> 9 25.58/8.95 mark1(9) -> 3 25.58/8.95 f1(0, 0) -> 10 25.58/8.95 mark1(10) -> 4 25.58/8.95 a1() -> 11 25.58/8.95 ok1(11) -> 5 25.58/8.95 b1() -> 12 25.58/8.95 ok1(12) -> 5 25.58/8.95 h1(0) -> 13 25.58/8.95 ok1(13) -> 2 25.58/8.95 g1(0, 0) -> 14 25.58/8.95 ok1(14) -> 3 25.58/8.95 f1(0, 0) -> 15 25.58/8.95 ok1(15) -> 4 25.58/8.95 proper1(0) -> 16 25.58/8.95 top1(16) -> 6 25.58/8.95 active1(0) -> 17 25.58/8.95 top1(17) -> 6 25.58/8.95 mark1(7) -> 17 25.58/8.95 mark1(8) -> 8 25.58/8.95 mark1(8) -> 13 25.58/8.95 mark1(9) -> 9 25.58/8.95 mark1(9) -> 14 25.58/8.95 mark1(10) -> 10 25.58/8.95 mark1(10) -> 15 25.58/8.95 ok1(11) -> 16 25.58/8.95 ok1(12) -> 16 25.58/8.95 ok1(13) -> 8 25.58/8.95 ok1(13) -> 13 25.58/8.95 ok1(14) -> 9 25.58/8.95 ok1(14) -> 14 25.58/8.95 ok1(15) -> 10 25.58/8.95 ok1(15) -> 15 25.58/8.95 proper2(7) -> 18 25.58/8.95 top2(18) -> 6 25.58/8.95 active2(11) -> 19 25.58/8.95 top2(19) -> 6 25.58/8.95 active2(12) -> 19 25.58/8.95 b2() -> 20 25.58/8.95 mark2(20) -> 19 25.58/8.95 b2() -> 21 25.58/8.95 ok2(21) -> 18 25.58/8.95 proper3(20) -> 22 25.58/8.95 top3(22) -> 6 25.58/8.95 active3(21) -> 23 25.58/8.95 top3(23) -> 6 25.58/8.95 b3() -> 24 25.58/8.95 ok3(24) -> 22 25.58/8.95 active4(24) -> 25 25.58/8.95 top4(25) -> 6 25.58/8.95 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (6) 25.58/8.95 BOUNDS(1, n^1) 25.58/8.95 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 25.58/8.95 Transformed a relative TRS into a decreasing-loop problem. 25.58/8.95 ---------------------------------------- 25.58/8.95 25.58/8.95 (8) 25.58/8.95 Obligation: 25.58/8.95 Analyzing the following TRS for decreasing loops: 25.58/8.95 25.58/8.95 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.58/8.95 25.58/8.95 25.58/8.95 The TRS R consists of the following rules: 25.58/8.95 25.58/8.95 active(h(X)) -> mark(g(X, X)) 25.58/8.95 active(g(a, X)) -> mark(f(b, X)) 25.58/8.95 active(f(X, X)) -> mark(h(a)) 25.58/8.95 active(a) -> mark(b) 25.58/8.95 active(h(X)) -> h(active(X)) 25.58/8.95 active(g(X1, X2)) -> g(active(X1), X2) 25.58/8.95 active(f(X1, X2)) -> f(active(X1), X2) 25.58/8.95 h(mark(X)) -> mark(h(X)) 25.58/8.95 g(mark(X1), X2) -> mark(g(X1, X2)) 25.58/8.95 f(mark(X1), X2) -> mark(f(X1, X2)) 25.58/8.95 proper(h(X)) -> h(proper(X)) 25.58/8.95 proper(g(X1, X2)) -> g(proper(X1), proper(X2)) 25.58/8.95 proper(a) -> ok(a) 25.58/8.95 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 25.58/8.95 proper(b) -> ok(b) 25.58/8.95 h(ok(X)) -> ok(h(X)) 25.58/8.95 g(ok(X1), ok(X2)) -> ok(g(X1, X2)) 25.58/8.95 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 25.58/8.95 top(mark(X)) -> top(proper(X)) 25.58/8.96 top(ok(X)) -> top(active(X)) 25.58/8.96 25.58/8.96 S is empty. 25.58/8.96 Rewrite Strategy: FULL 25.58/8.96 ---------------------------------------- 25.58/8.96 25.58/8.96 (9) DecreasingLoopProof (LOWER BOUND(ID)) 25.58/8.96 The following loop(s) give(s) rise to the lower bound Omega(n^1): 25.58/8.96 25.58/8.96 The rewrite sequence 25.58/8.96 25.58/8.96 g(mark(X1), X2) ->^+ mark(g(X1, X2)) 25.58/8.96 25.58/8.96 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 25.58/8.96 25.58/8.96 The pumping substitution is [X1 / mark(X1)]. 25.58/8.96 25.58/8.96 The result substitution is [ ]. 25.58/8.96 25.58/8.96 25.58/8.96 25.58/8.96 25.58/8.96 ---------------------------------------- 25.58/8.96 25.58/8.96 (10) 25.58/8.96 Complex Obligation (BEST) 25.58/8.96 25.58/8.96 ---------------------------------------- 25.58/8.96 25.58/8.96 (11) 25.58/8.96 Obligation: 25.58/8.96 Proved the lower bound n^1 for the following obligation: 25.58/8.96 25.58/8.96 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.58/8.96 25.58/8.96 25.58/8.96 The TRS R consists of the following rules: 25.58/8.96 25.58/8.96 active(h(X)) -> mark(g(X, X)) 25.58/8.96 active(g(a, X)) -> mark(f(b, X)) 25.58/8.96 active(f(X, X)) -> mark(h(a)) 25.58/8.96 active(a) -> mark(b) 25.58/8.96 active(h(X)) -> h(active(X)) 25.58/8.96 active(g(X1, X2)) -> g(active(X1), X2) 25.58/8.96 active(f(X1, X2)) -> f(active(X1), X2) 25.58/8.96 h(mark(X)) -> mark(h(X)) 25.58/8.96 g(mark(X1), X2) -> mark(g(X1, X2)) 25.58/8.96 f(mark(X1), X2) -> mark(f(X1, X2)) 25.58/8.96 proper(h(X)) -> h(proper(X)) 25.58/8.96 proper(g(X1, X2)) -> g(proper(X1), proper(X2)) 25.58/8.96 proper(a) -> ok(a) 25.58/8.96 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 25.58/8.96 proper(b) -> ok(b) 25.58/8.96 h(ok(X)) -> ok(h(X)) 25.58/8.96 g(ok(X1), ok(X2)) -> ok(g(X1, X2)) 25.58/8.96 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 25.58/8.96 top(mark(X)) -> top(proper(X)) 25.58/8.96 top(ok(X)) -> top(active(X)) 25.58/8.96 25.58/8.96 S is empty. 25.58/8.96 Rewrite Strategy: FULL 25.58/8.96 ---------------------------------------- 25.58/8.96 25.58/8.96 (12) LowerBoundPropagationProof (FINISHED) 25.58/8.96 Propagated lower bound. 25.58/8.96 ---------------------------------------- 25.58/8.96 25.58/8.96 (13) 25.58/8.96 BOUNDS(n^1, INF) 25.58/8.96 25.58/8.96 ---------------------------------------- 25.58/8.96 25.58/8.96 (14) 25.58/8.96 Obligation: 25.58/8.96 Analyzing the following TRS for decreasing loops: 25.58/8.96 25.58/8.96 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). 25.58/8.96 25.58/8.96 25.58/8.96 The TRS R consists of the following rules: 25.58/8.96 25.58/8.96 active(h(X)) -> mark(g(X, X)) 25.58/8.96 active(g(a, X)) -> mark(f(b, X)) 25.58/8.96 active(f(X, X)) -> mark(h(a)) 25.58/8.96 active(a) -> mark(b) 25.58/8.96 active(h(X)) -> h(active(X)) 25.58/8.96 active(g(X1, X2)) -> g(active(X1), X2) 25.58/8.96 active(f(X1, X2)) -> f(active(X1), X2) 25.58/8.96 h(mark(X)) -> mark(h(X)) 25.58/8.96 g(mark(X1), X2) -> mark(g(X1, X2)) 25.58/8.96 f(mark(X1), X2) -> mark(f(X1, X2)) 25.58/8.96 proper(h(X)) -> h(proper(X)) 25.58/8.96 proper(g(X1, X2)) -> g(proper(X1), proper(X2)) 25.58/8.96 proper(a) -> ok(a) 25.58/8.96 proper(f(X1, X2)) -> f(proper(X1), proper(X2)) 25.58/8.96 proper(b) -> ok(b) 25.58/8.96 h(ok(X)) -> ok(h(X)) 25.58/8.96 g(ok(X1), ok(X2)) -> ok(g(X1, X2)) 25.58/8.96 f(ok(X1), ok(X2)) -> ok(f(X1, X2)) 25.58/8.96 top(mark(X)) -> top(proper(X)) 25.58/8.96 top(ok(X)) -> top(active(X)) 25.58/8.96 25.58/8.96 S is empty. 25.58/8.96 Rewrite Strategy: FULL 25.97/9.41 EOF