/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxTRS (1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxTRS (3) CpxTrsMatchBoundsProof [FINISHED, 12 ms] (4) BOUNDS(1, n^1) (5) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (6) TRS for Loop Detection (7) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(c) -> mark(f(g(c))) active(f(g(X))) -> mark(g(X)) proper(c) -> ok(c) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: active(c) -> mark(f(g(c))) active(f(g(X))) -> mark(g(X)) proper(c) -> ok(c) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) CpxTrsMatchBoundsProof (FINISHED) A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 6. The certificate found is represented by the following graph. "[26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63] {(26,27,[active_1|0, proper_1|0, f_1|0, g_1|0, top_1|0]), (26,28,[mark_1|1]), (26,31,[ok_1|1]), (26,32,[ok_1|1]), (26,33,[ok_1|1]), (26,34,[top_1|1]), (26,35,[top_1|1]), (26,36,[top_1|2]), (26,37,[top_1|2]), (26,42,[top_1|3]), (26,50,[top_1|3]), (26,54,[top_1|4]), (26,55,[top_1|4]), (26,58,[top_1|5]), (26,60,[top_1|5]), (26,63,[top_1|6]), (27,27,[c|0, mark_1|0, ok_1|0]), (28,29,[f_1|1]), (29,30,[g_1|1]), (30,27,[c|1]), (31,27,[c|1]), (32,27,[f_1|1]), (32,32,[ok_1|1]), (33,27,[g_1|1]), (33,33,[ok_1|1]), (34,27,[proper_1|1]), (34,31,[ok_1|1]), (35,27,[active_1|1]), (35,28,[mark_1|1]), (36,31,[active_1|2]), (36,38,[mark_1|2]), (37,28,[proper_1|2]), (37,41,[f_1|2]), (37,48,[ok_1|3]), (38,39,[f_1|2]), (39,40,[g_1|2]), (40,27,[c|2]), (41,29,[proper_1|2]), (41,43,[g_1|2]), (41,46,[ok_1|3]), (42,38,[proper_1|3]), (42,44,[f_1|3]), (42,52,[ok_1|4]), (43,30,[proper_1|2]), (43,45,[ok_1|2]), (44,39,[proper_1|3]), (44,47,[g_1|3]), (44,51,[ok_1|4]), (45,27,[c|2]), (46,45,[g_1|3]), (47,40,[proper_1|3]), (47,49,[ok_1|3]), (48,46,[f_1|3]), (49,27,[c|3]), (50,48,[active_1|3]), (50,53,[mark_1|4]), (51,49,[g_1|4]), (52,51,[f_1|4]), (53,45,[g_1|4]), (54,52,[active_1|4]), (54,56,[mark_1|5]), (55,53,[proper_1|4]), (55,57,[g_1|5]), (55,51,[ok_1|4]), (56,49,[g_1|5]), (57,45,[proper_1|5]), (57,49,[ok_1|3]), (58,56,[proper_1|5]), (58,59,[g_1|6]), (58,62,[ok_1|5]), (59,49,[proper_1|6]), (59,61,[ok_1|4]), (60,51,[active_1|5]), (61,27,[c|4]), (62,61,[g_1|5]), (63,62,[active_1|6])}" ---------------------------------------- (4) BOUNDS(1, n^1) ---------------------------------------- (5) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (6) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(c) -> mark(f(g(c))) active(f(g(X))) -> mark(g(X)) proper(c) -> ok(c) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (7) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence g(ok(X)) ->^+ ok(g(X)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / ok(X)]. The result substitution is [ ]. ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(c) -> mark(f(g(c))) active(f(g(X))) -> mark(g(X)) proper(c) -> ok(c) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: active(c) -> mark(f(g(c))) active(f(g(X))) -> mark(g(X)) proper(c) -> ok(c) proper(f(X)) -> f(proper(X)) proper(g(X)) -> g(proper(X)) f(ok(X)) -> ok(f(X)) g(ok(X)) -> ok(g(X)) top(mark(X)) -> top(proper(X)) top(ok(X)) -> top(active(X)) S is empty. Rewrite Strategy: FULL